The 19-year-old, who had scored in each of his two previous domestic games, came off the bench for Marcus Rashford after 72 minutes in north London.
Greenwood first laid on a first-time cross for Edinson Cavani to put United ahead. Then he helped United finish with a flourish by firing past Hugo Lloris to cap off the scoring with virtually the last kick of the game.
According to Opta data, Greenwood is the first United teenager to score and produce an assist in the Premier League since Cristiano Ronaldo, during his first campaign with the Reds in 2003/04.
Greenwood has also now moved level with Rashford as United's second-highest teenaged scorer in the Premier League, with 13 goals.