Leeds goalkeeper Paddy Kenny made a superb early double save, first keeping out Ross Wallace's 20-yard shot and then blocking Dean Marney's follow-up.
The visitors had to wait until the 13th minute for their first effort when El-Hadji Diouf's shot from inside the penalty area was deflected for a corner.
Burnley striker Martin Paterson lashed a shot from the edge of the area straight at Kenny in the 29th minute, while Diouf spurned a gilt-edged chance at the other end 60 seconds later, pulling his shot wide from 12 yards.
Kenny dealt comfortably with McCann's shot from outside the area and Wallace's long-range free-kick early in the second half, and Leeds responded through Rodolph Austin, who fired in an angled shot from 20 yards.
Leeds were enjoying their best spell of the game when winger Aidy White forced Burnley goalkeeper Lee Grant to spill the ball with a venomous shot from the edge of the area.
Leeds' Austin tested Grant again from outside the box in the 68th minute.
Burnley striker Austin finally broke the deadlock in the 83rd minute, heading sweetly into the bottom corner from Kieran Trippier's fine right-wing cross.
Leeds defender Jason Pearce went close to snatching an equaliser from close range in the 89th minute following Brown's free-kick, but Burnley held on to register their first win over Leeds in six matches.