Stuart Fleetwood's first-half strike gave Hereford three vital points at the expense of Crewe.
Clayton Donaldson and Danny Shelley both forced Bulls keeper Adam Bartlett into early action as the visitors started the brighter of the two teams.
Fleetwood struck for the 16th time this season after latching on to Kenny Lunt's through ball.
Bartlett made further crucial saves from Shelley and Joel Grant to preserve the hosts' slender advantage.
The Bulls' win lifts them nine points clear of trouble, going into two away matches in four days at Accrington and Gillingham.
Crewe are now seven points adrift of the play-off pack ahead of next Saturday's home date with Cheltenham.
Hereford manager Jamie Pitman told BBC Hereford and Worcester:
"While I don't like to pick too many out, of the back lads, Michael Townsend in particular was excellent as was Stefan Stam.
"From that you can build and get your chances, then if you take them you have a chance of winning the game.
"The crowd got behind us well today and it gave the team a massive lift. I don't think people always realise how important that is to make sure we get something from games."
Crewe manager Dario Gradi told BBC Radio Stoke:
"I didn't think they were better than us. It was a poor game lacking in atmosphere.
"If we win seven games, I think we'll be in the play-offs and we'll go until it's not possible.
"Next Saturday I'll pick the best team I can pick."