El-Hadji Diouf was Leeds's biggest threat in the first half, and his vicious strike in the 13th minute might have drawn first blood had it not been blocked.
Burnley could have snatched a lead just before the half-hour mark. Martin Paterson found space to shoot, but failed to trouble Kenny.
The Clarets created the first clear-cut chance of the second period through Chris McCann, who gave his marker the slip only to be denied by Kenny.
Wallace was a live wire all evening, testing Kenny on more than one occasion in the second half, but Leeds came close themselves on 63 minutes. Aidan White's shot was parried by home keeper Lee Grant, and Burnley cleared the danger.
Rodolph Austin tested Grant with a volley in the 68th minute, and chances went begging for Paterson and Austin at the other end.
Burnley snatched a winner, but they left it late. McCann swung in a direct free kick in the 83rd minute and Austin leaped highest and found the back of the net to send Leeds home empty-handed.