Pogba cancelled out Patrick Bamford's opener then struck the winner before Anthony Kay picked up two yellow cards in quick succession for a foul and dissent.
The visitors went in front with 31 minutes gone when former Crewe striker Ryan Lowe crossed for Bamford to power home a superb header.
Dean Bowditch saw a shot saved by Alex keeper Steve Phillips, Darren Potter's rebound hit Mark Ellis and Ryan Lowe blazed over at the third attempt as the Dons further pressed.
But Crewe equalised five minutes before the break when Pogba nodded in Gregor Robertson's cross at the far post.
After the restart, Dons midfielder Chris Lines hit the post with a dipping volley and Bowditch was called offside as he rattled the woodwork from the follow-up.
Shaun Williams almost restored the visitors' lead with 10 minutes to go but he cracked the crossbar with a free-kick from the edge of the box.
But Pogba had the last word, burying the winner at the second attempt after Ian McLoughlin saved his initial shot with six minutes to go before Kay got his marching orders.
- Sports & Recreation
- Patrick Bamford
- Ryan Lowe
- Dean Bowditch