Albion came from behind to fight their way through to the second round of the Worthington Cup.
Barry McConnell gave third division toilers Exeter a shock fifth minute lead with a penalty conceded by Guy Butters.
Shaun Wilkinson, deputising for Paul Kitson, struck a spectacular equaliser on 34 minutes.
The Seagulls dominated the rest of the tie, but they had to wait until the 13th minute of extra time to edge ahead.
Captain Danny Cullip, fresh from signing a new three-year contract, clinched their first Withdean win of the season with a courageous header.
Albion made one change to the team beaten 1-0 at Millwall on Saturday. Paul Kitson was rested because of his back problem, so Wilkinson returned to the starting line-up.
Wilkinson joined Marney in an inexperienced front partnership, as manager Martin Hinshelwood switched to his preferred 4-4-2 formation, with Steve Melton on the right side of midfield.
Exeter, defeated 3-1 at home by Bournemouth on Saturday, made a couple of changes. Curran came in for Alex Watson and Walker replaced the injured Steve Flack.
A minute's silence was observed before kick-off on the first anniversary of the terrorist atrocities in New York. The memorial flag for one of the victims, Albion fan Robert Eaton, was passed over the heads of spectators in the South Stand and then paraded on the pitch.
Albion were given a warning as early as the fifth minute. Gaia's cross was nodded down by Goodman for former Seagull Martin Thomas to strike a shot from 20 yards, which Petterson fisted away.
Albion fell behind moments later. Butters, making his home debut, was controversially deemed to have fouled Walker and McConnell sent Petterson the wrong way from the resulting penalty.
The unmarked Curran then met a corner with a thumping header which Petterson, fresh from agreeing another month's contract, did well to parry.
The stunned Seagulls responded to their unexpected deficit by carving out a couple of good chances of their own.
Cullip's header from a free kick released Wilkinson at the far post. His first senior goal beckoned, but Miller made a smothering stop at the expense of a corner.
Wilkinson then delivered a teasing cross from the right flank which Marney glanced over the bar. Butters almost made amends for his spot kick misery in the 19th minute with a firm downward header from a Paul Watson corner, which Miller managed to keep out with an outstretched boot.
Albion were almost caught out by a rapid counter-attack from the lively visitors.
Young Irish midfielder Cronin, a target for Wigan, freed Coppinger on the right wing and his low cross was just out of the reach of the lunging Goodman at the far post.
The Seagulls needed to retain their composure because the Exeter defence was not exactly looking water-tight either.
Albion's frustration was emphasised by a booking for Melton on 32 minutes for a foul on Coppinger near the half-way line.
Wilkinson ended Albion's Withdean famine in glorious fashion eleven minutes from the break.
He controlled a cross from Watson and turned to fire the sweetest of right foot shots from 20 yards which the stationary Miller could only admire.
It was a wonderful way for Wilkinson to break his duck and it ended a home drought for the Seagulls dating back to the departed Lee Steele's last gasp winner against Bristol City at the end of last season.
More importantly it was precisely the sort of confidence boost Albion required.
Exeter suffered a blow just before the interval when the evergreen Goodman hobbled off. He was replaced up front by Australian Harries, signed on Tuesday by the West Countrymen on a non-contract basis.
Albion made a substitution for the second half, introducing Nathan Jones for Paul Brooker in a straight swap on the left.
Wilkinson and Marney were serving the Seagulls well in the absence of Kitson, the injured Bobby Zamora and lone signing Graham Barrett, prevented from playing by Arsenal.
Wilkinson nearly struck again on 56 minutes, screwing a shot narrowly wide of the far post after a penetrating cross from Melton eluded everybody in the box.
The Seagulls went close again when another Butters' header from a Watson corner was inadvertently helped onto the roof of the net by Marney.
Albion were in complete control of the tie in terms of possession at this stage.
Their domination was not being reflected by the score, as Exeter, to their credit, continued to defend stoutly.
The Seagulls' persistent pressure was almost rewarded in the 85th minute. Wilkinson rounded keeper Miller from Melton's pass, but he was forced wide in the process and Curran blocked at the expense of a corner.
Curran was cautioned for dissent on the stroke of normal time. The ensuing free kick was advanced ten yards by referee Joslin and Richard Carpenter curled it just wide of the far post from the edge of the area.
Cullip gave Albion the lead they craved with two minutes left in the first period of extra time. His firm glancing header from an Charlie Oatway free kick found the far corner of the net.
It was a typically brave effort from the Seagulls' determined skipper and he was quickly back in the action after treatment.
- Albion line-up (4-4-2): Petterson (gk) 7, Watson (lb) 6, Cullip (cd) 9, Butters (cd) 6, Oatway (m) 6, Brooker (lw) 6, Carpenter (m) 6, Pethick (rb) 7, Wilkinson (f) 8, Marney (f) 8, Melton (rw) 7. Subs: Rogers, Jones 6 for Brooker (withdrawn 46), Packham, McPhee for Marney (withdrawn 111), Hammond.
- Exeter (3-5-2): Miller, Power, Walker, Curran, Roscoe, McConnell, Gaia, Thomas, Goodman, Coppinger, Cronin. Subs: Whitworth, Ampadu for Walker (withdrawn 96), Harries for Goodman (injured, 43), Breslan for Thomas (withdrawn 108), Fraser.
- Scorers: Wilkinson (34), Cullip (103). Bookings: Melton (32).
- Attendance: 5,200.
- Fan's View, by Paul Gunn, Hove "After a nervous start and going a goal down, Brighton began to put a decent performance together, albeit against lower league opposition.
"The two young lads, Wilkinson and Marney, ran their socks off and Wilkinson capped his performance with a brilliantly taken goal. A fit Bobby Zamora would have had a hatful.
"Brighton need to carry on and this win will hopefully restore confidence. Let's hope Rocketman keeps sending them up on Saturday.
"And, with a Premiership draw in round two, who knows? It could be Beckham at Withdean."
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