After Rodrigo's deflected strike put Leeds ahead inside fifteen minutes, the Whites squandered bags of strong chances to extend their lead.
Kenny McLean looked to have rescued a point for the visitors with moments left to play as he poked home a Teemu Pukki cross, but 19-year-old substitute Joe Gelhardt had the final say in injury time.
The victory took the fifth-bottom Whites four points clear of the Premier League drop zone with nine games left, although second-bottom Burnley who are five points adrift have two games in hand.
Graham Smyth, the YEP's chief football writer, hands out the scores from Sunday afternoon's epic clash at Elland Road.
7. Mateusz Klich
8 - Linked the play really well from back to front, covered a huge distance. Picture by Tony Johnson.
8. Raphinha
8 - Deserved a goal for his performance. Gave Norwich a torrid time as the game opened up and hit the bar twice. Assisted the winner. Photo by PAUL ELLIS/AFP via Getty Images.
9. Dan James
7 - Decision making in the final third aside he put in a good shift. Ran himself into the ground. Photo by George Wood/Getty Images.
10. Rodrigo
8 - A big improvement from his last outing. Besides the goal there was some lovely play. Showed fight too. Photo by George Wood/Getty Images.
11. Patrick Bamford
7 - Made a difference to the attack, showed little rust until he missed his big chance. Withdrawn at the break. Photo by George Wood/Getty Images.
12. Jack Harrison (sub)
6 - Showed fight, evidently has been listening to his new head coach, but could have picked up a second yellow. Photo by Michael Regan/Getty Images.