Albion edged through to Sunday's F.A. Cup Second Round draw.
Bobby Zamora was the match winner yet again for the Seagulls netting his 14th goal of the season in the first half.
Peter Taylor's team had to fight all the way for their 16th win at Withdean out of 17 against their spirited Third Division opponents.
They were never at their fluent best, but Albion will be relieved to have avoided an upset.
It does not matter what the competition is, Zamora just keeps on scoring.
He struck stylishly on 31 minutes, to the relief of a 5,450 crowd.
Zamora, latching on to Charlie Oatway's clever chip, calmly rounded keeper Ian Dunbavin to slot home for his fifth game running.
The Seagulls, with Robbie Pethick replacing the unwell Gary Hart, were in command until Shrewsbury, 30 places below them in the Football League, rallied in the last half hour.
Little Luke Rodgers, their own 13-goal marksman, shot just wide and then appealed for a penalty when sent sprawling as keeper Michel Kuipers raced off his line in the 71st minute.
Referee David Crick awarded Albion a free-kick instead and booked Rodgers for diving.
Danny Cullip, transfer-listed at his own request and making his 100th Albion appearance, helped the Seagulls to their third clean sheet in succession.
Shrewsbury finished with ten men, Leon Drysdale seeing red with four minutes left for a second cautionable foul on Zamora.
Table-topping Albion, now unbeaten in seven matches under Taylor, are scheduled to return to Falmer for training on Monday ahead of Wednesday's home game against Peterborough.
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