Francois Zoko and Yoann Arquin both scored on their league debuts as 10-man Notts County ended Crewe's 20-game unbeaten run.
Zoko scored from close range and Arquin netted the visitor's second only to see red two minutes later and County held on after Mathias Pogba scored his second goal in two games as Crewe threatened a comeback.
Alan Sheehan hit the Alex upright with a deep free-kick, but County made their dominance count five minutes before the break when Julian Kelly fired in an inviting cross and Zoko got in front of Harry Davis to volley home from six yards out.
Crewe should have levelled after the restart but Sheehan did well to clear Max Clayton's side-footed effort off the line.
County doubled their lead in the 58th minute when Alan Judge centred for Arquin to score from 12 yards, but they suffered a setback two minutes later when the striker saw red for a late challenge on Ashley Westwood.
Pogba headed in Westwood's corner at the far post to give Crewe a lifeline in the 73rd minute and it took a last-ditch clearance from Lee Hughes to keep Westwood from levelling at the death.