Dario Gradi's League One outfit deserved their passage into the third round draw, which was sealed when teenager Nicky Maynard added a second in the final minute.
Paul Jewell rested a number of his frontline stars, but still named a formidable Latics line-up that should have been strong enough to progress at Gresty Road.
But Wigan's Swedish striker Andreas Johansson set the tone for the night when he scooped an easy chance over in the seventh minute.
The visitors dominated possession in the first half but failed to test home keeper Ben Williams.
Former Crewe defender David Wright went close with a shot across goal, and a Matt Jackson header was flying in until Luke Varney got his head in the way to divert the ball over.
But Jack, who has only just returned to Crewe on a shot-term contract, provided the breakthrough three minutes before the break. Varney swept past Fitz Hall and pulled the ball back from the byline and Jack got in first at the near post to fire the ball home.
Crewe carried their attacking momentum into the second half and Varney raced onto David Vaughan's pass only for Mike Pollitt to make a vital save.
Wigan edged back into contention as Jewell began to ring the changes, bringing on Arjan De Zeeuw and then Emile Heskey, but Williams proved the hero for Crewe with fine saves from Leighton Baines and Heskey.
Crewe were not just prepared to sit back and defend and pushed forward in search of a clincher.
Maynard twice had chances but shot wide and over when in on goal. But when Varney cut the ball back in the dying seconds the teenager made no mistake.