Charlie Austin headed his 20th goal of the season late on as the Clarets registered their first win against Leeds United in six attempts.
Former Swindon striker Austin, who has scored in all but three of his 17 appearances for Burnley this season, broke the deadlock in the 83rd minute when he headed home defender Kieran Trippier’s cross.
Dyche, installed as Eddie Howe’s successor at Turf Moor last week, saw his new side canter to victory over Wolves at the weekend and inspired them to a fourth win in their last five league games thanks to the prolific Austin.
Burnley leapfrogged the visitors into 10th place in the table, while Leeds, without a win in their last five league matches, have slipped into the bottom half of the table.
Burnley midfielder Chris McCann started in place of defender David Edgar, while Leeds recalled right-back Adam Drury and 11-goal leading scorer Luciano Becchio.
Leeds goalkeeper Paddy Kenny made a superb early double save, first keeping out Ross Wallace’s 20-yard shot at full stretch and then blocking Dean Marney’s follow-up, while the visitors had to wait until the 13th minute for their first effort on goal 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.
Burnley threatened twice in quick succession to break the deadlock towards the end of the first half.
Defender Kieran Trippier’s powerful drive forced Kenny into a fumbling save and from the subsequent corner Austin fired his shot from inside the area straight at Kenny.
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 before Burnley defender Ben Mee was forced off injured and replaced by Danny Lafferty in the 59th minute.
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, with midfielder Michael Brown’s effort blocked by home defender Jason Shackell soon after.
Austin tested Grant again from outside the box in the 68th minute and finally broke the deadlock in the 83rd minute, heading sweetly into the bottom corner from 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.