Sako was released by Crystal Palace in the summer but had found his chances limited under Darren Moore.
The Mali international took his rare chance to start with the only goal just after the half hour in front of a crowd of just 15,465.
West Brom had won 2-0 when the sides met in the league at the same venue on Boxing Day and both teams were much changed from their last outings. Moore made nine changes to his line-up and Wigan counterpart Paul Cook seven.
West Brom goalkeeper Jonathan Bond kept the scores level after 10 minutes when he dived to his left to keep out a low shot from Will Grigg at the expense of a corner. Wigan's Chey Dunkley headed over from the flag kick.
Josh Windass tried his luck with a shot from 20 yards which Albion cleared as Wigan continued to boss the early stages.
Chris Brunt curled over a free-kick from the left as West Brom finally threatened but Kyle Bartley's header from the edge of the six-yard box at the back post was blocked.
Just before the half hour Brunt curled a free-kick towards the top right corner of the Wigan goal but Jamie Jones stretched to tip it over for a corner.
Wigan failed to properly clear Brunt's corner from the left and when the West Brom captain curled the ball back in, Sako's header went in off the bar for his first Baggies goal with 31 minutes on the clock.
Mason Holgate could have marked his debut with a goal just before half-time but, after cutting in from the right and evading two challenges, the on-loan Everton man's shot was comfortable for Jones to hold low to his right.
Sako should have doubled his side's advantage eight minutes into the second half but after being sent clear his shot was blocked by the legs of Jones before the striker's follow up was cleared on the line by Dunkley.
Jonathan Leko was the next West Brom player to race clear but his right-foot shot from the left side of the area went across goal and well wide of the target.
Holgate headed wide after Bartley had headed Brunt's corner back across goal as the home side pushed for the second goal their superiority deserved.
With Sako deployed on both wings, West Brom were bossing proceedings without finding the killer touch but they had already done enough to secure passage into the fourth round.