Signed by unpopular former manager Neil Warnock, Varney has spent large parts of this season as a terrace boo boy but his contribution here, cancelling out Jermaine Johnson's opener, has gone a long way to repairing that, although Wednesday may want to review his first which was given by a linesman as it came down off the bar.
The victory - Leeds' first in eight - vindicates their decision to bring in former Reading boss McDermott now instead of the planned date of the summer and the 52-year-old can thank Varney and Ross McCormack - architect of both goals - for the dream start.
Regardless of the enthusiasm generated by his appointment, though, McDermott was manning what had been a rapidly sinking ship and that showed in the early stages, despite Leeds' best efforts.
Steve Morison wasted a golden 10th-minute chance when he mis-kicked in front of a near-open goal, while Rudy Austin spurned a number of other good opportunities with wild shots from distance.
At that point Wednesday had still to amount an attack of note but all that changed when the took the lead.
Chris Kirkland's clearing kick was allowed to bounce by Jason Pearce and, with centre-back partner Stephen Warnock caught ball-watching, Johnson nipped in between him and Kenny and coolly lobbed the advancing keeper.
Minutes later they thought they had a penalty when Stuart Holden clipped Adam Drury in the box but, despite replays suggesting the offence had taken place a yard inside, a free-kick was given on the edge and McCormack's shot was cleared off the line by Miguel Llera.
Pearce then wasted a free header as Leeds tried to find a way back and they could have been doing it against 10 men had referee Neil Swarbrick not taken a lenient view of the already-booked Giles Coke's trip on Paul Green.
Coke was wisely subbed off at the break and the Owls received another let off five minutes after the restart when Morison, clean through after a Llera slip, dallied and eventually fell over as Reda Johnson chased him down.
Leeds drew level in the 64th minute, though, Varney connecting with a McCormack cross and having his header given as goal by the linesman as it bounced down off the bar.
McCormack's cross made the goal all the easier to score and three minutes later the pair linked up again to devastating effect. McCormack went on a run not too dis-similar to the one Sergio Aguero went on against Manchester United on Monday but, rather than shooting, he stood up a cross for Varney who did the rest.
Having gone into their shell after their goal, Wednesday had to wake up and they did; Steve Howard cracking the post with a volley and Paddy Kenny producing a brilliant to stop in the midst of a scramble, before Leroy Lita dragged wide when well set.