Charlie Austin took his tally for the season to 20 goals with a second-half equaliser and an extra-time winner to leave Harry Redknapp’s side potentially 90 minutes from a return to the Premier League.
Wigan, who were held 0-0 at home in the first leg, had led from the ninth minute when James Perch prodded in a cross from James McClean.
Rangers squandered chance after chance to level but were thrown a lifeline in the 72nd minute when Junior Hoilett won a penalty which Austin converted.
And six minutes into extra-time the former Burnley frontman struck again to book a winner-takes-all showdown with Derby at the end of the month.
Yet Wigan had stunned the hosts by taking an early lead when Perch darted in front of Rangers skipper Clint Hill to get to McClean’s centre first and poke the ball home.
Clearly a man for the big occasion, the goal was only Perch’s second of the season – the other being the winner against Manchester City in the FA Cup which did send Wigan to Wembley.
McClean could have doubled the lead when he latched onto Gary O’Neil’s woeful backpass but instead of rounding the advancing Rob Green, he blatantly dived over the Rangers keeper and got a deserved yellow card.
Green then did well to keep out a low drive from Shaun Maloney but, as the first half wore on, Rangers gradually took control.
However, they were continually frustrated as Scott Carson palmed away Niko Kranjcar’s fierce drive and Austin’s header was cleared off the line by McClean.
In the second half McClean had a chance to wrap the game up for Wigan when he charged through again, this time staying on his feet, but having over-run the ball he could only find the outside of the post from an acute angle.
Instead the hosts’ pressure finally told as Hoilett attempted a one-two with substitute Bobby Zamora on the edge of the area, only for Gary Caldwell to halt the winger’s run with a clumsy challenge allowing Austin to step up and equalise.
Zamora almost won it in normal time when he held off two Wigan defenders before lobbing onto the roof of the net.
But the former England international had a hand in the winner when he lifted the ball into Austin’s path and watched his strike partner tuck it past Carson.
Rangers survived a late scare when Rob Kiernan’s shot was deflected inches wide in stoppage time and they held out to secure a passage to Wembley and keep their promotion dream alive.