Barnsley 0 Sunderland 2
Sunderland were shut out by the hard-working Yorkshire strugglers but finally found the breakthrough when Leadbitter - in for the injured Carlos Edwards - struck after 66 minutes. Connolly added the second a minute from time.
The Black Cats, backed by more than 8,000 fanatical fans, had two surprise absentees from their squad - Anthony Stokes and Tobias Hysen - rumoured to have been left in Sunderland after arriving late for the team coach.
But Keane's side were anxious to pull themselves closer to second-placed Derby County and carved out a string of first-half chances.
Leadbitter struck a dipping 25-yard free-kick but it gave Nick Colgan a comfortable save, and when Connolly cut inside his 20-yard shot was well over the top.
Liam Miller missed a good opportunity to open the scoring after a gap had opened up in the Barnsley defence, and when Stern John finally had a shot on target it was blocked by Brian Howard.
A shot on the run from Ross Wallace was then arrowing goalwards before Paul Heckingbottom cleared the effort in front of keeper Colgan.
Connolly produced two shots in as many minutes but his first was over the top after he had twisted and turned to create the opening and Colgan held onto the second.
But Barnsley - who were pinned back for much of the opening 45 minutes - had the best chance of the half after 25 minutes when Sam Togwell's cross was headed on by Howard and Hungary international Peter Rajczi hooked his shot just wide from eight yards.
Leadbitter and Connolly created two more efforts that were well-blocked by more brave interventions from the home side.
A minute before the break John got the better of Paul Reid and the ball broke for Dean Whitehead racing through, but as Colgan tore off his line he fired wide of the target.
Istvan Ferenczi then worked hard to find a gap for the Tykes but his shot was blocked by Darren Ward.
But Sunderland continued to have the lion's share of possession leaving the home side to work overtime at the back and another opportunity fell to Whitehead who lashed a shot side from 18 yards.
Barnsley then had strong appeals for a penalty turned down when Howard's header appeared to be handled by Jonny Evans.
The visitors finally broke Barnsley's stubborn resistance in the 66th minute with a rare shot on target. Leadbitter collected the ball outside the penalty area before smashing home a 22-yard angled shot.
Tykes keeper Colgan appeared to have a good view of the ball but was deceived when it bounced before it reached him and skidded into the net.
Seconds before the final whistle Connolly was sent forwards by a towering Evans header before lashing home a low shot from the edge of the area.