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.
1. Illan Meslier
6 - Poor pass led to a corner from which Norwich scored. Didn't have a huge lot to do besides that.
2. Diego Llorente
6 - Lost his man for the goal, wasn't up to the standard he's set for himself but the effort was there. Defended aggressively.
3. Pascal Struijk
7 - Another unfussy display. Nice run and pass to put Leeds on the front foot for the first goal.
4. Liam Cooper
7 - Lots of defending to do but did it with a minimum of fuss.
5. Kalvin Phillips
7 - A performance more about graft than quality. Got stuck in, put himself about. Not at his best but solid for the vast majority.
6. Jamie Shackleton
6 - Busy. Booked, taking one for the team. Leeds' main problems didn't come down his side.