The victory lifted Manchester United to 21 points to move up to sixth, while Tottenham have 20 points in eighth.
Daniel James started on the right side in the 22nd minute and then sent a cross to Lingard in the penalty area, paving the way for Greenwood from behind, and he fired a powerful globe taken by Gazzaniga into the corner.
Manchester regained control of the game again, to launch Rashford rocket from outside the penalty area in the 25th minute, collided with the Argentine goalkeeper.
Manchester United clinched the Old Trafford tie with Tottenham Hotspur on Wednesday night, defeating Spurs (2-1) in the 15th round of the Premier League.
United's Marcus Rashford scored in two minutes (6: 49), while Tottenham's only goal was Dale Alli (39).
Manchester United translated early control of the game, with a first goal in the sixth minute, through a powerful ground shot to Rashford from the left side of the penalty area, collided with goalkeeper Gazanega and the post and netting.
Spurs scored for the equalizer, and in the 12th minute Spurs got a foul on the edge of the area.
In the 18th minute, United also get a foul from the edge of the penalty area, and Rashford hit a powerful ball, which passed a little near the post.