The Glovers took the lead after just 51 seconds when James Hayter's neat chest down found Joe Edwards who drilled past Steve Phillips from the edge of the box.
Steve Davis' side nearly equalised in the 10th minute when Byron Moore volleyed at goal, but his effort was tipped over the bar by Stech.
The hosts lost both captain Byron Webster and Nathan Ralph to injury in the opening 25 minutes, but Yeovil nearly added a second when Hayter volleyed into the side netting from Sam Foley's cross.
Crewe tried to make the most of Yeovil's injury-hit back four but found Stech in imperious form as he denied both Max Clayton and Matt Tootle in the opening 45 minutes.
Alex, who are now without a win in four games, almost equalised through Chuks Aneke's glancing header but Stech was there again to ensure Yeovil earned all three points.