Cowie's first goal since September on the stroke of half-time and Connolly's second-half header proved the difference as the visitors ended a three-match winless run that had threatened to derail their automatic promotion hopes.
It was Wednesday, under the guidance of Dave Jones - a man who led the visitors to two play-off campaigns and an FA Cup final in his six years at the helm, who started the brighter.
Jeremy Helan cut inside and fired in a low shot from the edge of the area which David Marshall clutched.
At the other end, Fraizer Campbell's menacing run into the box forced a timely block out of Miguel Llera.
From Craig Bellamy's subsequent corner, Chris Kirkland fumbled the ball under pressure but was awarded a fortunate free-kick for some shoving in the area.
Owls striker Leroy Lita and referee James Linington took centre stage moments later when Cowie fell to the ground holding his face, with his team-mates claiming Lita had aimed an elbow at him. After allowing play to go on, the official returned to the incident and decided no further action was necessary.
But just as it looked as though it would be honours even at the break, Malky Mackay's side broke the deadlock.
The hosts failed to properly clear a corner and Cowie seized on the loose ball, struck his shot low and got a fortunate deflection off Anthony Gardner as the ball skidded into the back of Kirkland's net.
It was a bitter pill for Jones and his charges to swallow and they were almost caught napping early in the second half when Heidar Helguson looked to play Campbell in, but Miguel Angel Llera was alert to the danger and slid in brilliantly to thwart the striker.
There was no-one to save the hosts in the 65th minute when Peter Whittingham, making his 300th Cardiff appearance, arrowed in a free-kick from deep which Connolly nodded in from six yards for his fifth of the season.
Lita had an effort ruled out for offside and the visitors survived a goalmouth scramble late on, with just enough time left for Bellamy to endear himself to the home fans by going in the book for a crunching challenge on Johnson.