Paul Pogba added the third in the second half with his first goal of an injury-affected season.
United stayed fifth but moved within a point of fourth-placed Leicester, whose long stay in the top four could soon be over.
It might not matter. Fifth place will be enough to qualify automatically for the Champions League this season unless second-placed Manchester City get their two-year European ban overturned in the courts. A decision on City's appeal should be announced next week.
Tottenham, Champions League finalists last season, can forget about playing in Europe's elite competition next season and might not even make the second-tier Europa League.
Mourinho's team are ninth, nine points behind United, ahead of a North London derby against Arsenal on Monday morning.
United ran out comfortable winners at Villa Park but only after being set on their way by a controversially-awarded penalty. Fernandes went to ground just inside the area even though he appeared to tread on the right ankle of Villa defender Ezri Konsa after spinning on the ball.
After a consultation with VAR, the on-field decision to award a penalty was upheld and Fernandes sent Pepe Reina the wrong way from the spot in the 27th minute. The penalty was United's 13th of the season. No team has ever been awarded more in a single Premier League campaign.
Eighteen-year-old Greenwood has shown he is equally adept at shooting with his left or right foot after bursting on to the scene this season. His latest goal was with his right, with both feet off the ground as he fired home a rising shot from the edge of the area in first-half stoppage time.
Pogba and Fernandes are starting to click for United in midfield and they combined for the third goal. Fernandes found Pogba free at the edge of the area from a corner, and he curled home a finish past a stationary Reina.
Mourinho lamented the failure of the referee to award a penalty early in Spurs' game against Bournemouth for a push in the back on Harry Kane.
Maybe it was a ruse to hide the shortcomings of his team as Bournemouth faced no shots on target in a game for the first time since promotion to the Premier League in 2015.
Tottenham were somewhat fortunate to escape with a point, with Callum Wilson's goal from an overhead kick in the 90th minute ruled out after the VAR spotted the ball grazing the arm of fellow forward Josh King on its way into the net.
Bournemouth stayed in the relegation zone in third-to-last place but are now three points from safety with four games remaining.
Southampton's Danny Ings kept himself in the running to win the Golden Boot with his 19th goal of the season in a 1-1 draw at Everton. The striker is three goals behind leading scorer Jamie Vardy of Leicester.
Ings scored in the 31st minute, three minutes after James Ward-Prowse missed a penalty for the visitors, and Richarlison equalised shortly before halftime.
- AP
