After a tense opening quarter - understandable with so much at stake - Latics started to get going and began to create chances.
Fit-again James McClean put in a cross-shot which missed the far post by a whisker, also evading desperate lunges from Jack Whatmough and Callum Lang.
Max Power then attempted a free-kick from distance that went through the hands of former Wigan Academy goalkeeper Marko Marosi, who got just about enough on it to divert it behind for a corner.
But Power was on hand to provide the crucial opening goal two minutes before half-time.
The midfielder curled in a free-kick from the left-hand side that struck the near post, with Marosi beaten.Unfortunately for Shrewsbury, the ball rebounded into the unsuspecting Josh Vela, and back over the line into his own net.
That was the boost Latics needed, and they never looked back.
Power was involved in the second goal five after the restart that doubled the visitors' advantage.
His shot from the edge of the box clearly struck the hand of a home defender.
Will Keane stepped up to convert from 12 yards, his 26th goal of the campaign.
And the former Manchester United junior got his head to a James McClean free-kick on 65 minutes to seal a famous win - and kick-off a party that will go on for much of the summer.