The result keeps the Robins in touch with the teams just above the relegation zone, but leaves Crewe six points adrift at the foot of the table.
Central defender Drissa Diallo - a loan signing from Milton Keynes Dons - became Cheltenham 's 19th different goal scorer this season when he struck from a corner three minutes into the second half.
It was a tough afternoon for Crewe , who suffered a double calamity in the opening minutes when first James Bailey and then Calvin Zola were forced off injured. Bailey appeared to have a cut head, but Zola was carried off on a stretcher and taken to hospital for a check-up.
Both sides adopted a bold attacking policy with three forwards right from the start and it made for an open, attacking game.
Crewe should have taken advantage of some slack defending in the opening stages when Tom Pope, on for the injured Zola, fired over the crossbar then sent a free header straight at goalkeeper Scott Brown.
Cheltenham, however, dominated the spells either side of half-time and visiting goalkeeper Stuart Tomlinson was forced to make sharp saves from Scott Murray, Ashley Vincent and Barry Hayles. Vincent also placed a shot wide from a good position just inside the area.
The only goal came on 48 minutes when Diallo stole in front of his marker to meet a Josh Payne corner with a volley that flew inside the near post.
Darren Kenton's diving header almost made it 2-0 for Cheltenham, but first Colin Daniel then Shaun Miller missed good chances to get Crewe back into the game, firing wide after getting into good positions to shoot.
Crewe captain Julian Baudet almost forced a goal from a late goalmouth scramble, but Tomlinson was forced into a late save from Murray as Cheltenham played out the closing stages protecting their lead.