Michael Jackson's men knew that only a victory could take their relegation battle to the last weekend, but it was the visitors who came away with the points and in the process confirmed their spot in the play-offs.
Shrewsbury were given hope when Jon Taylor opened the scoring in the sixth minute, but Darren Ferguson's side were 3-1 ahead inside half an hour thanks to Conor Washington's double and a Lloyd Isgrove strike.
Tom Bradshaw reduced the deficit shortly before half-time, but the home side were hit on the break when Josh McQuoid scored in second-half injury time to end Shrewbury's two-year stay in the third tier.
The visitors' misery was further compounded by the red cards shown to Troy Archibald-Henville and Nathan Thompson in an ill-tempered finish that ensured Swindon's slim play-off chances were finally extinguished.
In other matches, Rotherham United - who will contest the play-offs - earned a dramatic 2-2 draw at home to MK Dons thanks to late goals from Michael O'Connor and Alex Revell, the visitors finishing with 10 men when Brendan Galloway saw red, while Preston North End are also building momentum ahead of the play-offs with a 3-1 victory over Gillingham at Deepdale.
