Purse faced being dropped after the embarrassing defeat but, with four players sidelined, the ex-Cardiff centre-back kept his place and provided an early breakthrough.
Wednesday went ahead on four minutes with a textbook goal when midfielder Darren Potter's teasing corner was headed in by Purse from close range.
They should have had a second after winger Michael McIndoe blundered and casually tapped a tame back-pass towards Keiren Westwood.
Luke Varney intercepted and blindly drilled the ball across the face of the open goal beyond the reach of Leon Clarke.
The Midlands side were in trouble again in the 20th minute as Wednesday grabbed a second goal after midfielder James O'Connor was sent sprawling by centre back Elliott Ward.
Up stepped Clarke to slam the spot-kick beyond Westwood's reach.
The Sheffield side had a let off in first-half stoppage time when Leon Best's header from a McIndoe free-kick appeared to be over the line with Grant clutching the ball but, with referee Colin Webster unsighted, Wednesday escaped.
Ward almost pulled one back with a spectacular overhead kick which was snatched on the line by Grant, before Coventry were seething again as Webster turned down their penalty appeals after Best went down in the area under a Mark Beevers challenge.
The Sheffield side eased off in the dying minutes as Coventry looked for a way back and on-loan Chelsea midfielder Jack Cork found the side-netting with a left-foot strike before being foiled by Grant.