The ex-Wolves striker found the target in the 43rd minute, but Wednesday had to cling on in the dying minutes as their South Yorkshire neighbours carved out a hatful of chances and missed the lot.
Doncaster Rovers had forced the pace in the opening minutes with centre-back Sam Hird foiled by Lee Grant as the keeper tipped his close-range header over the bar.
Full-back James O'Connor picked out Richie Wellens with his low strike deflecting off Mark Beevers, before Matt Mills steered a glancing header inches wide of the target.
The Hillsborough side almost broke the deadlock after neat work by Marcus Tudgay and Sean McAllister let in O'Connor, but his stinging drive from 25 yards was tipped over the target by Neil Sullivan.
The veteran keeper pushed away a long-range effort from Clarke and kept out Tudgay's low strike as O'Connor tidied up before Doncaster fell behind.
Dutch winger Etienne Esajas curled in a teasing free-kick and Clarke lunged forward to steer his header beyond the former Leeds keeper.
Wednesday escaped in the 57th minute when they failed to cut out a Gareth Roberts free kick as Hird stretched but pulled the ball wide at the far post.
Clarke should have set up Wednesday's second as he charged down the right flank but, with Tudgay and substitute Jermaine Johnson waiting, he miscued his cross as the ball curled out for a throw in.
Doncaster brought on Lewis Guy and James Hayter as the tried to snatch a point.
Midfielder John Spicer went close before Hayter almost levelled matters, but his low drive was stopped in acrobatic fashion by Grant as Wednesday clung on in a nail-biting finale.