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The Wigan keeper had to remain alert throughout the first half as Emyr Huws sent a well-placed cross towards the away goal, but Carson was able to tip it behind.
Jordi Gomez tried to extend Wigan's lead before the break, but Randolph made sure that his side were just a goal down at half time.
After a scrappy start to the second half, McManaman found himself through on goal looking for his second of the evening, but slipped as he tried to shoot, giving the hosts an easy clearance.
The home side, who occupy the final relegation place in the table, continued to apply pressure as they desperately looked to ease their fears of dropping into League One, but they were unable to find the back of the net before the end of the match.
The result means that Wigan join Derby County and Queens Park Rangers in the playoffs, while Birmingham will need one of the sides above them to slip up to avoid the drop at the weekend.