The Robins took an early lead through Alfie May before Massimo Luongo cancelled that opener out with an equaliser on seven minutes.
The home side then scored three second-half goals, including Jack Hunt's first Owls goal in 146 matches over two spells, to get back to winning ways after back-to-back draws.
Wednesday now have a two-point gap on seventh-placed Sunderland.
Story of the match
It was an early scare for the Owls, as an unchallenged free-kick was met by the head of Matty Blair, who forced a smart save out of Bailey Peacock-Farrell down to his right. But the linesman had raised his flag for offside.
However, Peacock-Farrell was left hapless on four minutes as May saw his effort from Elliott Bonds’ pass loop over the Northern Ireland international via a fortunate deflection to hand the Robins the lead.
The Owls did not dwell on the early goal, as they levelled the game three minutes later. A Barry Bannan corner was flicked towards goal by Harlee Dean, and with his back to goal, Luongo superbly hooked the ball over the shoulder of Owen Evans and into the back of the net.
On 11 minutes, the returning Nathaniel Mendez-Laing sharply cut into the area from the left-hand side and bent an effort that was held by Evans.
Despite upping the ante after the early setback, Darren Moore’s side lacked that little bit of quality in the final third and saw many chances be wasted thanks to them overhitting crosses or misplacing passes.
On the half-hour mark, Lee Gregory won a freekick on the left-hand side of the area. Instead of delivering it into the area, Bannan cut the ball back to an unmarked Marvin Johnson, who struck a first-time effort through a crowd of bodies, but Evans managed to make the block on the line.
As the half came to a close, Mendez-Laing forced Evans into a phenomenal point-blank save, but he was flagged for being in an offside position to latch onto the end of a sublime ball from Bannan.
The second period began as it ended, with the home side in control and in search of the advantage.
But, again, Wednesday showed carelessness in possession; for example, Mendez-Laing was slipped down the right-hand side, but instead of delivering the ball into the centre early for Gregory, he hesitated, and his subsequent cross was headed clear.
However, Wednesday found the goal they desired on the hour through Gregory, whose first touch set him up for a long-distance effort that he struck sweetly into the bottom corner.
Cheltenham should have equalised on 64 minutes. The Robins worked the ball into the box with May squaring the ball to George Lloyd. The pass, however, was slightly behind the forward, and by the time he gained control and had the shot, Peacock-Farrell rushed out and made a great block before Wednesday cleared.
However, whilst Wednesday only had a one-goal advantage there was the chance for the Robins to snatch one back. Bonds struck the upright from 25 yards before May was denied on the slide by the recovering Dean.
But the Owls sealed their place in the top six with two goals in four minutes, as Hunt scored his first goal for the club with a sweet volley at the back stick before Byers found the target after jinking past his marker inside the area.
Sheffield Wednesday: Peacock-Farrell; Storey (Iorfa, 83'), Dean, Hutchinson; Hunt, Byers, Luongo, Bannan, Johnson; Mendez-Laing (Sow, 65’), Gregory (Paterson, 89')
Unused Subs: Wildsmith, Dele-Bashiru, Kamberi, Dunkley
Cheltenham Town: Evans; Raglan (Sercombe, 75’), Pollock, Boyle; Long, Wright, Bonds, Chapman (Etete, 78’), Hutchinson; May, Blair (Lloyd, 62’)
Unused Subs: Flinders, Freestone, Thomas, Williams