Callum McManaman's close-range finish with just 10 minutes left gives Roberto Martinez side a crucial three points that closes the gap on the sides just above them in the table.
Shane Long gave the Baggies the lead from close range only for Arouna Kone to head the visitors level just before the break.
The home side re-took the lead five minutes into the second half when Gareth McAuley headed home Graham Dorrans' free-kick before Wigan substitute James McArthur brought the visitors level again eight minutes later with his first touch.
Despite McManaman's late intervention giving Wigan all three points, they stay in the bottom three, though they are now just two points behind 17th placed Sunderland.
In a quiet opening 20 minutes Latics forward Kone had the only real chance, his fizzing drive from distance smartly pushed away by Ben Foster.
The deadlock was broken just before the half hour when, after Paul Scharner lost the ball in midfield, Romelu Lukuku played a fine pass forward to Markus Rosenberg, whose low cross was converted at close range by Long.
Wigan were staring down the barrel at that stage but nearly equalised on 36 minutes when Shaun Maloney's free-kick had Foster beaten but curled just wide of the right-hand post.
Two minutes later though the Latics did equalise, Jean Beausejour whipping in a wonderful cross from the left onto the head of Kone, whose looping effort beat a motionless Foster.
The Baggies found themselves back in front five minutes into the second half when Dorrans' free-kick found the unmarked McAuley, who sent a downward header beyond Joel Robles.
Wigan boss Martinez threw on McArthur just before the hour and was rewarded instantly when the midfielder converted a teasing cross from Maloney with a smart diving header.
The Latics then took the lead for the first time with just 10 minutes left when McManaman sidefooted home from close range after another jinking run by Maloney down the left.
There was still time for late drama when McArthur cleared off his own goalline in injury-time, but Wigan clung on for the win.