A first-half strike by Adrian Forbes gave the Swans the victory they needed and now automatic promotion rests on the result next Saturday.
Swansea started brightly in front of a packed Vetch Field for the last time in a league match. But it was the Shrews who threatened first when a Gary Monk header fell to Kevin Street, but he blasted a first-time shot high and wide.
But the goal all Swansea was waiting for came in the eighth minute when a cross from Sam Ricketts was partly cleared by Trevor Challis only for Kevin McLeod to snap up the loose ball. He crossed to the unmarked Adrian Forbes, who steadied himself before driving into the back of the net.
Shrewsbury almost silenced the huge crowd five minutes later when Willy Gueret had to go full length to save at the feet of Kelvin Langmead after a run and cross from Sam Aiston.
Gueret came to Swansea's rescue once again when he pushed away a low shot from the busy Kelvin Langmead as play switched from end to end.
Sam Aiston was proving to be a danger down the visitors' left and another deep cross from the former Sunderland man was met by Langmead, but his firm header was beaten away by yet another superb stop from the inspired Gueret in the home goal.
Midway through the second half the strangely-quiet Lee Trundle was given a clear shooting chance by McLeod, but he blazed high and wide from a central position. Minutes later it was Kristian O'Leary who sent the ball into the stand as a cross fell his way.
The Swans just couldn't find that second goal to put the spirited Shrews away, and Lee Thorpe joined the fray to give extra power up front.
Monk headed over his own crossbar as Swansea's nerves were jangling in the final minutes, but once again Gueret came to the rescue by punching the resulting corner clear, leaving that single Forbes strike enough to take the promotion issue right to the wire.