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.