Paulo Sousa's side, fresh from a 1-0 victory over Championship leaders Wolves, over-ran the Sheffield side for most of the game, but could find no way past the ex-Derby keeper as Clarke struck with 15 minutes to go.
Grant's one-man show began when he foiled midfielder Hogan Ephraim, but he blundered minutes later when his poor clearance went straight to centre-back Mark Beevers five yards outside of the Wednesday box.
Patrick Agyemang moved in quickly to rob the lanky defender as he raced into the area before unselfishly squaring to Heldar Helguson as Grant spread himself to deflect the striker's effort at the foot of the post.
The Sheffield side were in deep trouble in the 27th minute when Helguson was sent sprawling by Frank Simek as he chased a Peter Ramage cross, but once again Grant was in the way blocking skipper Martin Rowlands' fierce spot-kick.
Wednesday, on the back foot for long periods, were finally handed a half-chance as Richard Wood rose unchallenged to meet a James O'Connor corner, but his tame header was easily snapped up by Radek Cerny.
Rangers were on top but were caught out after Francis Jeffers had a shot blocked. The ball fell nicely for Clarke, who slammed it straight at Cerny from 10 yards.
Agyemang showed his pace in the 52nd minute racing down the right flank but, as Grant charged out, the midfielder pulled his right-foot strike wide of the far post from 16 yards.
Wednesday were caught out again in a carbon copy move as Helguson outpaced his marker to cut inside before lashing in a low strike which was pushed out by the Wednesday keeper.
The Sheffield side struck in the 75 minute when midfielder Marcus Tudgay lashed a low drive which Cerny could only push away and Clarke steamed in to grab his fifth of the season.