Albion, who had never met Shrewsbury in the FA Cup, started brightly as they looked to ensure there were no slip ups against their Third Division opponents.
But it was the Shrews who had the first shot of the match, Jemson hitting an optimistic effort from 30 yards. It looked to be going wide, but Kuipers took no chances and saved comfortably by his right hand post.
Albion's dead-ball specialist Watson, who has scored for the Seagulls in the first round in the last two seasons, went for the hat-trick with an inswinging free-kick from the right wing which Dunbavin did well to push over the bar under pressure from Zamora.
Watson then picked out Albion utility-man Robbie Pethick who whipped in a cross to Steele lurking in the box, but Heathcote made a crucial interception to deny the front man the opportunity to break his FA Cup duck.
Casual defending almost cost Albion in the 13th minute, but Shrewsbury spurned the chance to take a surprise lead.
Drysdale's throw was headed by Jemson into the path of Moss, who fired wide of the far post with his left foot from 15 yards.
Jemson and Moss were both given too much space in that instance by Albion's normally miserly back four.
Pethick was booked in the 20th minute for a foul on Atkins ten yards inside Albion territory. The resulting free-kick from Shrewsbury's former Blackburn midfielder was headed away by Zamora.
Steele, requiring a couple more goals to reach his half century, had a shot on the turn from outside the area from a Carpenter pass which bounced straight to Dunbavin.
Albion carved out a good opening midway through the half. Steele spread the play to Jones and his deep cross was headed over from beyond the far post by the usually lethal Zamora.
But the Albion hitman doesn't spurn too many chances and he duly gave the home side the lead on 31 minutes.
Oatway's clever lob released him deep inside the area and he rounded keeper Dunbavin before slotting into the empty net from a tight angle.
Zamora's 51st goal for the club followed his decisive contributions in Albion's 1-0 victories in the last two home League games against Port Vale and Colchester.
The unstoppable 20-year-old has also now netted in five appearances in a row.
Very little had been seen of Rodgers for Shrewsbury so far, although he had the chance to equalise almost instantly.
The ball reached him at an awkward height ten yards out and his poorly struck volley offered Kuipers a routine stop.
Drysdale had his name taken for a foul on Zamora close to halfway.
Shrewsbury had done their best to frustrate the Seagulls, quickly funnelling eight men behind the ball when they were not in possession, but Zamora had yet again given Albion a relatively firm grip on the game.
Half-time: Albion 1, Shrewsbury 0.
Atkins came close for Shrewsbury seconds into the restart with a well-struck drive from 25 yards.
Unfortunately for the visitors it swerved the wrong way, narrowly wide of the far post.
A low cross-shot by Jones for the Seagulls moments later lacked the power to bother Dunbavin.
Albion put together one of their best moves of the match early in the second half. Carpenter and Pethick combined well to release Watson whose cross was just out of Steele's reach at the back of the box.
There was little to suggest at this stage that Shrewsbury were capable of an equaliser, although Albion wanted a second goal to give them security.
Zamora almost provided it on 56 minutes when he steered a header wide after Steele had set up Mayo to cross.
Zamora was unmarked and the Shrewsbury defence appealed in vain for offside.
A misplaced pass by Jemson inside the Albion half allowed Pethick to launch a rapid counter-attack with a pass to Zamora, scampering down the right flank.
His crowd-pleasing back-heel released Watson to cross. Heathcote partially cleared and Zamora's follow-up header from long range was caught by Dunbavin beneath the crossbar.
Shrewsbury made a substitution with an hour gone, Jenkins replacing Rioch at leftback.
Albion received a warning that the tie was far from over three minutes later when Rodgers, racing on to Jagielka's header, flashed an angled drive past the diving Kuipers and just wide of the far post.
Morgan protested that Albion's big Dutch keeper should have come off his line to eradicate the rare moment of danger.
Albion made a change on the left flank midway through the second half, introducing Brooker for Jones on the left wing. Meanwhile Shrewsbury brought on Aiston for Moss.
Shrewsbury thought they should have been awarded a penalty when Rodgers, sprinting on to another Jagielka header, was sent sprawling as Kuipers rushed off his line.
Instead referee Crick awarded Albion a free-kick and booked Rodgers for diving.
Shrewsbury were reduced to ten men with four minutes remaining when Drysdale received a second yellow card for another foul on Zamora.
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