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.
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.
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.
