Kenny Jackett's men went into Saturday's game knowing that a win would secure promotion after Rotherham United could only manage a 0-0 draw at home to Bradford City on Friday.
They were ahead on the stroke of half-time when Kevin McDonald's deflected effort beat goalkeeper Ben Garratt and relegation-threatened Crewe's task was made even harder when Chuks Aneke saw red for lashing out at Richard Stearman just before the break.
Dave Edwards wrapped up the win midway through the second half to allow the celebrations to begin.
Second-placed Brentford slipped to a 1-0 loss at Swindon Town to hand Wolves a six-point advantage in the title race, Louis Thompson scoring for the play-off chasing hosts and Adam Forshaw seeing red for Mark Warburton's side.
Leyton Orient are seven points behind Warburton's men after a brace from David Moody inspired Russell Slade's charges to a thumping 5-1 win at home to Gillingham.
Preston North End, already assured of a play-off place, also dished out a hammering as they beat struggling Carlisle United 6-1 at Deepdale, with Craig Davies netting a second-half hat-trick.
The Lancashire side are likely to be joined in the play-offs by Peterborough United, who secured a narrow 1-0 win over Coventry City, while MK Dons' faint hopes of a top-six finish have been all but extinguished following a 2-0 home loss to Crawley Town.
At the other end of the table, Notts County gave their bid for League One survival a huge boost with a 4-2 win over Port Vale, while bottom-of-the-table Stevenage are four points from safety after a 3-2 defeat at the hands of fellow relegation candidates Colchester United.
Elsewhere, Tranmere Rovers, fresh from parting company with manager Ronnie Moore, eased their own relegation fears with a 2-1 over struggling Shrewsbury Town, and Bristol City are practically safe after beating Walsall 1-0.
