Adam Virgo emerged as the unlikely hero as Albion snatched a place in the play-off final at Cardiff from the jaws of despair.
Sam Parkin's 81st minute strike for Swindon, his 23rd of the season and fourth against the Seagulls, wiped out the first leg advantage.
Substitute Rory Fallon then left Mark McGhee's men staring an agonising exit in the face with a second goal for the visitors seven minutes into extra time.
But Virgo forced penalties with a dramatic diving header following a throw-in by Charlie Oatway with just seconds of the extra period remaining.
Albion made the most of the reprieve, even though regular penalty taker Leon Knight had gone off injured.
Richard Carpenter, Chris Iwelumo, substitute John Piercy and the outstanding Virgo all scored from the spot.
Tommy Mooney, who converted a penalty for Stoke at Withdean last season, had his effort saved by Ben Roberts and when Andy Gurney struck the post Albion were through to a money-spinning clash with Bristol City on May 30.
The capacity of the play-offs to provide the unexpected surfaced even before the kick-off with a dramatic change in weather conditions and a surprise in the Swindon line-up.
A violent thunderstorm had ground staff forking and mopping a pristine looking Withdean pitch.
While Albion were unchanged, visiting manager Andy King gave Grant Smith, son of former Seagull Gordon, his first League start for the club at the expense of Andrew Nicholas.
The fit-again Stefani Miglioranzi resumed in midfield for Swindon in place of David Duke, who was injured in Albion's 1-0 win in the first leg at the County Ground on Sunday.
Former jockey Willie Carson, the Swindon chairman, made the short journey from the meeting at Goodwood in the hope of watching his team "dump on the Seagulls."
They were marginally quicker out of the starting stalls in a predictably edgy and frenzied opening, with so much at stake.
Roberts' long, unbeaten sequence for the Seagulls was threatened on a couple of occasions. The keeper had to smother at Parkin's feet from Matthew Hewlett's pass. The ball then squirmed out of his grasp for a worrying moment before it was cleared.
The other sniff at goal for Swindon arose from a rare mistake by Guy Butters, which allowed Mooney to hook a shot straight at Roberts at his near post.
Albion fans collectively held their breath on the stroke of half-time when a foul by Oatway on Hewlett gave Swindon a free-kick just outside the area. Gurney's ensuing attempt deflected off the head of Iwelumo in the defensive wall.
Albion hearts fluttered again from the corner which followed. It was struck deep to the far post by Igoe on to the head of Parkin and then Heywood. His header goalwards hit Virgo and ricocheted conveniently into the hands of Roberts.
One of the biggest cheers of the evening was reserved for the appearance during the interval of Albion goalscoring legend Peter Ward, who had flown over from the States to watch the game.
There had not been much of the kind of flair Ward used to display in the opening 45 minutes between two very evenly matched sides.
Smith came perilously close to fluking a lead for Swindon three minutes into the restart when his free-kick from the right hand touchline, delivered with his left foot, curled over Roberts and inches wide of the far post.
The near miss was immediately followed by a substitution by McGhee, Hart replacing Reid on the right just as he had in the second half on Sunday.
Albion suffered a blow with the loss of 26-goal Knight through injury. The little striker was caught late by Miglioranzi as a corner was cleared. He came back on briefly after treatment but had to be replaced by Hinshelwood.
Within five minutes of his introduction, Hinshelwood met a deep cross by Jones with an angled volley which forced Evans into his first serious save at the expense of a corner.
The closest escape yet for Albion came on 78 minutes when a left foot shot on the turn by Mooney from 20 yards struck the top of the bar.
Mooney had also been denied by the bar in the first leg on Sunday, but three minutes later his partner Parkin finally brought to an end Albion's proud defensive record.
An angled drive by Igoe was parried by Roberts. The ball came back across the box and, with Albion scrambling to clear, Parkin stabbed in Swindon's lifeline from close range.
The first goal conceded by Albion from more than ten and a half hours was followed by another seven minutes into extra time as Swindon turned the tie on its head.
Harding gave possession away inside his own half, Mooney capitalised with a clever pass through for the advancing Fallon to clip over Roberts.
Suddenly it was Albion who now needed something dramatic. Piercy, on for the extra period for Jones, nearly produced it with a low shot into the side netting from 20 yards.
When Iwelumo shot tamely at Evans and Hinshelwood headed over from Virgo's cross, all hope seemed to have evaporated for Albion.
Then up popped Virgo in the final minute of extra time to prolong an extraordinary evening into a penalty shoot-out.
The Seagulls simply refused to be denied. Virgo was among their scorers in the shoot-out as Albion clinched a place at Cardiff after being on the brink of despair.
ALBION (4-4-2): Roberts (gk); Cullip (cd), Knight (f), Oatway (cm), Carpenter (cm), Butters (cd), Jones (lm), Virgo (rb), Harding (lb), Iwelumo (f), Reid (rm). Subs: Kuipers, Mayo, Hart for Reid (withdrawn 48), Piercy for Jones (withdrawn 91), Hinshelwood for Knight (injured 66).
Scorer: Virgo (120).
Penalties: Carpenter, Iwelumo, Piercey, Virgo.
Bookings: None.
SWINDON TOWN (4-4-2): Evans; O'Hanlon, Smith, Hewlett, Gurney, Igoe, Mooney, Parkin, Miglioranzi, Howard, Heywood. Subs: Griemink, Nicholas for Smith (withdrawn 91), Fallon for Howard (withdrawn 72), Reeves for Igoe (injured 106), Pook.
Scorers: Parkin (81), Fallon (97).
Bookings: Hewlett (34, foul).
Attendance: 6,876.
Fans' View:
MICK WRIGHT (Crawley Down)
I thought Chippy (Richard Carpenter) had a corker of a game in midfield and he was my man of the match. All credit to Adam Virgo for scoring the goal in the dying seconds and having the bottle to put his penalty away. At least they can close the roof at the Millennium Stadium if it rains.
DAVE ROBERTS (Brighton)
We were second best again and McGhee's tactics were quite negative, not having a specialist striker like McPhee or Robinson on the bench. Having said all that, this is a massive result for the club and I hope we sell out our allocation for Cardiff.
JANE BOWER (Eastbourne)
What a brilliant night! In the cold light of day, we were quite poor going forward and predictably lacked creativity in midfield but who cares about that?
JEFF YATES (Swindon fan)
Obviously we are all totally gutted by that floodlight robbery. I don't think I'm being biased when I say we were by far the better team again, so to lose like that is a real kick in the teeth. Good luck to Brighton in the final as we all hate Bristol City.
Albion player ratings:
BEN ROBERTS 8: Albion's custodian made a few decent saves and was a little unfortunate with Parkin's aggregate equaliser. Gave the Seagulls the edge in the penalty shoot-out by saving from Mooney.
DANNY CULLIP 7: Another solid display by the captain against Swindon's prolific front pairing. It was his flick-on from a throw-in which created the goal.
LEON KNIGHT 6: The little marksman had no chances to end his mini drought. The injury he suffered midway through the second half did not help Albion's cause.
CHARLIE OATWAY 6: Typically dogged and determined in the middle of midfield. His throw-in led to Virgo's dramatic aggregate equaliser.
RICHARD CARPENTER 7: His goal in the first leg looked like being decisive for so long. Industrious as ever alongside Oatway and scored Albion's first penalty with aplomb.
GUY BUTTERS 7: Another sterling effort from the veteran centre half. Held the defence together again with Cullip.
NATHAN JONES 6: Not really a night for wingers. The Welshman, so desperate for a return to Cardiff, worked hard until he was taken off for extra time.
ADAM VIRGO 9: The young rightback was the pick of the bunch even before he scored. Very strong for his age and displayed big match temperament.
DANIEL HARDING 6: The experience gained by the leftback during the run-in will hold him in good stead. A better passer than he showed.
CHRIS IWELUMO 6: The big target man was well shackled by Heywood, but held his nerve in the shoot-out.
PAUL REID 6: The young Australian had little opportunity in a frantic first half to press his claims for a longer contract before Hart replaced him.
Substitutes GARY HART 6: Added a touch more zest to Albion's attacking play after replacing Reid early in the second half.
ADAM HINSHELWOOD 6: Demonstrated his adaptability by making his first appearance since December on the right side of midfield in place of the injured Knight.
JOHN PIERCY 6: Came close with a shot after replacing Jones for extra time. Calmly converted the third penalty.
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