Leo Fortune-West put the visitors ahead on eight minutes when he scored with a free header from Derek Asamoah's cross.
Asamoah doubled the lead when his shot was parried in by Wrexham goalkeeper John Ruddy and then home defender Steve Evans was sent off for an elbow.
Fortune-West made it three after the break from a low cross, Danny Williams then replying with a 30-yard cracker.
Shrewsbury midfielder Mike Jones found the back of the net with the last kick of the game, but the goal was ruled out because he had fired in from an indirect free-kick.
The result leaves Wrexham in the drop zone, but Shrewsbury move up three places to 10th in the table.
Wrexham: Ruddy, Danny Williams, Steve Evans, Gareth Evans, Roche, Whitley, Mark Jones, Garrett (Carvill 61), Barron (Mike Williams 53), Roberts (McEvilly 76), Llewellyn.
Subs Not Used: Michael Jones, Done.
Sent Off: Steve Evans (40).
Booked: Roberts, Danny Williams, Llewellyn, McEvilly.
Goals: Danny Williams 55.
Shrewsbury: MacKenzie, Hope (Mike Jones 56), Burton, Langmead, Herd, Edwards, Drummond, Davies, Tierney (Ashton 46), Asamoah (Hall 83), Fortune-West.
Subs Not Used: Shearer, Canoville.
Booked: Tierney, Edwards.
Goals: Fortune-West 8, Asamoah 38, Fortune-West 51.
Att: 5,605.
Ref: M Dean (Wirral).