Goals in each half by centre-half Richard Naylor and midfielder Ian Westlake gave Town a 2-0 lead, and although Lee Peacock's late reply made for a nervy ending, the hosts held out fairly comfortably.
Town started with four teenagers on the pitch, giving a debut to striker Dean McDonald and a home debut to goalkeeper Shane Supple, who came on at Leicester last week after first choice Lewis Price was injured.
After a quiet start, Town went ahead on seven minutes when a corner was headed out to Jim Magilton who floated the ball back in for Naylor to head home from 10 yards.
Wednesday went close to an equaliser four minutes later when Fabian Wilnis fouled David Graham on the edge of the area and Chris Brunt sent a curling free-kick into the side-netting.
Wednesday had another good chance on 21 minutes when a swift break ended with Chris Eagles crossing low for Burton O'Brien who scuffed wide from around the penalty spot.
Wilnis shot over when David Lucas fluffed his punch at a corner before Darren Currie tried his luck from 30 yards but only found the goalkeeper's body.
Wednesday boss Paul Sturrock clearly wasn't happy with his side's display and made a triple substitution on the hour, bringing on former Town player Drissa Diallo plus Peacock and Adam Proudlock for Steve Adams, Eagles and Graham.
It seemed to liven the visitors up with Diallo immediately getting in a header which Supple saved comfortably, before Proudlock headed a John Hills cross over when unmarked six yards out.
Brunt shot across goal before Wednesday's revival was hit by a second well-worked Town goal as Luis Castro Sito and Currie worked a one-two before the Spaniard pulled the ball back for Westlake to fire home on the turn.
That looked to have sealed the game, but Wednesday ensured a nervy finish when sub Peacock rose unchallenged to head home from around six yards after an accurate Glenn Whelan corner with nine minutes to go.