With fellow strugglers Derby losing at Preston, it was a good night for Wednesday in their battle to beat the drop.
Following a good counter-attack from the visitors, Adam Armstrong pulled the ball back to Sam Gallagher, who scuffed his shot wide.
A Harris cross picked out Windass, but his volley went well over.
Windass put the hosts ahead eight minutes before the break with a deflected shot after being set up by Harris.
Wednesday posed further problems for the opposition defence in the closing stages of the half, firstly when Barry Bannan's free-kick found Tom Lees, who volleyed over.
Wednesday carried on from where they left off at the start of the second half with Liam Palmer putting a shot wide and Bannan's free-kick picking out Julian Borner, who put his close-range header off-target.
When Corry Evans threatened to find a way through for the visitors, Lees timed his challenge inside the area to perfection to dispossess him.
As Wednesday went in search of a second goal, Palmer fired in a shot on goal which was saved by Kaminski.
Buckley failed to capitalise on a great chance to equalise, shooting straight at Westwood after Armstrong squared the ball to him inside the area.
A volley from Borner, following a Bannan corner, forced a good save from Kaminski.
With the Wednesday players looking to run the clock down, Armstrong tried a stoppage-time shot which was deflected wide.
If Wednesday can build on this result, their survival fight could yet go down to the wire with a trip to Derby on the final day of the season looming.