The Railwaymen moved off the bottom of the Championship table for the first time since December 31st with two first-half goals from on loan striker Gareth Taylor proving to be the difference.
And had it not been for some brilliant stops by Scott Carson the margin of victory would have been greater.
The Owls had the better opening and Glenn Whelan should really have done better when Jon Paul McGovern cut the ball back across the box but the Irish midfielder side-footed over the top.
Deon Burton flashed an angled drive into the side netting after moving on to a flick from Marcus Tudgay.
But Carson saved the visitors on 20 minutes with a brilliant reflex save to tip over a close-range Steve Jones volley.
However, he could do nothing to prevent the Cheshire side edging in front on the half hour.
After Whelan gave the ball away in the middle of the pitch, Luke Rodgers burst through and fed Billy Jones who surged to the byline before pulling the ball back for Taylor to stab in from close range.
Rodgers' pace was unsettling the Wednesday defence and young centre-half Patrick Collins was booked for bringing the former Shrewsbury man down.
Carson came up with another important save to push away a header from Madjid Bougherra after Kenny Lunt had pumped a corner into the box.
Crewe were finishing the half strongly, although Ross Turnbull needed to get down to save Burton O'Brien's low 25-yard drive.
But it was Crewe who got the psychological boost of a second goal in first-half stoppage time when Steve Jones delivered a great cross from the right and the unmarked Taylor raced in to nod home.
Wednesday pushed forward more after the break and Turnbull made a couple of important stops including an acrobatic dive to turn away substitute Lee Bullen's rising shot.
But Crewe could have had more goals. Steve Jones scuffed wide and then pulled another effort off target within the space of a couple of minutes and Taylor was denied his hat-trick by Carson just over the hour mark.
Shortly after that Crewe escaped when Bougherra tripped Burton in the box but referee Clive Penton waved away the penalty appeals.