Callum McManaman scored after just three minutes at St Andrew's to move Uwe Rosler's side four points clear of seventh-placed Brighton and Hove Albion with just one game remaining.
Wigan require just a point from their final match of the season - which comes at Blackburn Rovers on Saturday - to ensure they finish fifth, which would lead to a play-off clash with QPR.
On the other hand, a fifth straight Championship defeat for 22nd-placed Birmingham means they need a positive outcome at Bolton Wanderers and for other results to go their way this weekend if they are to avoid the drop to League One.
In Tuesday's other Championship fixture, Charlton Athletic secured survival with a 3-1 home win over mid-table Watford.
Callum Harriott opened the scoring for Jose Riga's Charlton on 22 minutes, before Troy Deeney levelled matters for the away side after an hour.
Charlton captain Johnnie Jackson restored their lead shortly after, though, before Harriott popped up with his second of the game 13 minutes from time to move the London club five points clear of Birmingham.
In League One, Carlisle United missed the chance to boost their hopes of avoiding relegation with a 0-0 draw at Crawley Town.
The stalemate means Carlisle are three points adrift of safety, and they need an unlikely win at champions Wolves on the final day to stand any hope of staying up.
Elsewhere, Chris Porter's late equaliser earned Sheffield United a 1-1 draw at Oldham Athletic.
