The 19-year-old saw his thumping opener after the break quickly cancelled out when James Tarkowski rose above Harry Maguire to level.
Greenwood put the hosts back in front six minutes from time with a fierce left-footed strike that deflected beyond goalkeeper Bailey Peacock-Farrell at the front post.
Edinson Cavani then tapped in from close range in stoppage time to wrap up the victory at Old Trafford.
The visitors’ direct attacking play had caused United problems of their own and they had the ball in the net after 14 seconds.
Chris Wood beat goalkeeper Dean Henderson to a floated delivery on the edge of the penalty area and headed into an empty goal, only to be correctly flagged offside.
It means Ole Gunnar Solskjaer’s side are now eight points behind leaders City in second place with six games to play, while Burnley remain 17th and six points above the relegation zone.
You can never be sure, said Greenwood about United’s hopes of catching their rivals. They might slip up, we have to concentrate on our next games.
Anything’s possible, hopefully they drop some points but we just have to keep focused on our games.