Gudjon Thordarson's side had to grind out victory in a dire spectacle at Gresty Road.
Chances were few and far between in both halves although Yeovil will have considered themselves to be the best side in the first 45.
They were imaginative in their build-up play but woeful around the box and were ultimately made to pay for two poor mistakes by keeper Josh Wagenaar.
In first-half stoppage time the keeper failed to gather a Crewe corner allowing Byron Moore a free shot on goal and Clayton Donaldson helped his co-striker's effort in at the far corner to give the home side a precious lead.
They rarely looked like letting that slip and, although Yeovil had plenty of possession in the second half, they lacked a cutting edge.
With 12 minutes to go Wagenaar was caught out again when Crewe's right wing-back John Brayford launched in a cross and the keeper stepped over the line in his desperate attempt to keep the ball out and spilled it into the net for Crewe's second goal.
Donaldson could have had a a second in the closing minutes but was denied by Wagenaar and substitute Steve Schumacher almost sneaked in on a Moore effort which the keeper pushed clear.
But Aaron Brown got back to hook the midfielder's effort off the line.