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.
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.