Brazil star Raphinha fired Leeds ahead in the 56th minute only for Andrew Omobamidele to head Norwich level from a corner two minutes later.
But Rodrigo's long-range strike put United back in front another two minutes later and the record signing's strike proved the winning goal against the division's bottom side.
The victory, United's second Premier League win of the season, took Marcelo Bielsa's side fourth-bottom and on to ten points after ten games played, three points clear of third-bottom Burnley.
Graham Smyth, the YEP's chief football writer, hands out the scores from the triumph at Carrow Road.
7. Stuart Dallas
7 - Lots of effort and running, tackled well, covered a lot of ground throughout. Came up big in the second half when needed. Picture by Nick Potts/PA Wire.
8. Raphinha
8 - Leeds' most dangerous player again. Gave them a first half outlet, scored the opener, threatened on the counter attack.
9. Dan James
6 - Struggled to make an impact. Lots of runs, tried to use his pace to stretch the defence. Nice pass out to Raphinha for the opener.
10. Rodrigo
7 - Couldn't create a great deal as Leeds struggled for cohesion but took his goal well and put a big shift in.
11. Jack Harrison
5 - Struggled to play as he can. Had a go up top in the central role and out wide, unable to shine in either. Always willing to get on the ball. Photo by Stu Forster/Getty Images.
12. Cody Drameh (sub)
6 - Went backwards a few times when the ball needed to go forward, a bit keen with a tackle that earned yellow but defended solidly.