Just short of the hour mark, Diomansy Kamara broke the deadlock of this constantly entertaining encounter to ultimately secure victory for the visitors, although they were made to work hard for their victory by a spirited and often enterprising Town outfit.
There was no doubting the importance Fulham boss Lawrie Sanchez placed on the fixture as he named his strongest available eleven, clearly keen to avoid a cup humbling against lower division opposition - managed by his former Wimbledon team-mate Gary Peters.
It was the Fulham defence, however, that looked humble in the second minute as young Shrewsbury winger Chris Humphrey came through two challenges to fire a low cross into a crowded goalmouth, where visiting goalkeeper Tony Warner diverted the ball to safety.
But from that point onward the next half hour was a succession of Fulham chances as Kamara, skipper Carlos Bocanegra and David Healy all went close to opening the scoring.
However, as the half came to close, on-loan Liverpool midfield man Jimmy Ryan sent a long ranger effort just over the Fulham crossbar, before warming the gloves of Warner with a crisp 20-yarder.
After dominating the early stages of the second period it was no surprise when the Craven Cottage outfit finally took the lead, Kamara climbing earliest and highest to score with a glancing header from Simon Davies' in-swinging corner.
But undeterred Shrewsbury battled back into the game and with six minutes remaining almost drew level, only for Darren Kempson's header to land in the side-netting - leaving Shrewsbury to take credit for a gutsy display, but Fulham with a place in round three.