Paul Sturrock's side were made to pay after carving out a hatful of chances in the opening half and squandering the lot.
Rowan Vine destroyed Wednesday in a four minute spell as Mike Newell's side chalked up their first away win for four months.
The 23-year-old striker was upended in the 52nd minute by Richard Wood and skipper Kevin Nicholls kept his cool to drive in the spot-kick.
And Vine, signed from Portsmouth in the summer for £250,000, killed off the struggling Hillsborough side on 56 minutes by smashing the ball into the roof of the net for Luton's first away win in four months.
Wednesday had failed to break down Luton in the opening half as they started with a flourish.
Marcus Tudgay went close twice in the opening minutes as the ex-Derby striker fired inches wide and headed over from close range.
Luton, with just one point from eight away games, hit back immediately when Steve Howard's low drive was snapped up by keeper David Lucas.
John Hills almost found a way through with a stinging shot from 25 yards before new signing Yoann Folly, on loan from Southampton for the rest of the season, picked out Deon Burton. His low strike was partly blocked by Markus Heikkinen and keeper Marlon Beresford snapped up the loose ball.
Midfielder Burton O'Brien went close when his ferocious drive flew over the bar as relegation threatened Wednesday staged a late rally.
Howard, a target for the Hillsborough side last season, almost added a third in the dying minutes as he connected with a Carlos Edwards cross but fired over from 12 yards.