The result left United second in the standings on 66 points from 32 games, eight behind leaders Manchester City and 10 ahead of third-placed Leicester City, who have a game in hand.
Burnley stayed 17th on 33 points from 32 games, six above 18th-placed Fulham who have played a game more and were held to a 1-1 draw at Arsenal after conceding a late equaliser in the lunchtime kickoff.
Greenwood was delighted and said that United were still in with an outside chance of clinching their first league title since 2013.
"I'm happy I'm playing a lot, (manager) Ole (Gunnar Solskjaer) trusts me and I do my best on the pitch and try to help my team mates and it's showing on the pitch," the 19-year-old forward told BT Sport.
"I keep focused, I knew my goals would come and kept my head down and I've taken my opportunities. Anything's possible, hopefully (Manchester City) drop some points but we just have to concentrate on our games."
Solskjaer heaped praise on Greenwood.
"Very happy for him," the Norwegian told the BBC. "He’s mixing up his game. He goes inside and outside, he's maturing all the time. It's lovely to see. He’s put the work in on the training ground and reaps the rewards."
DEFLECTED SHOT
Greenwood fired United ahead in the 48th minute and after James Tarkowski headed Burnley's equaliser two minutes later, he restored the home side's lead with a heavily deflected shot before Edinson Cavani sealed the win.
United enjoyed a huge let off in the first minute as Burnley striker Chris Wood headed the ball into the back of the net after only 13 seconds but his effort was ruled out for a marginal offside.
Visiting goalkeeper Bailey Peacock-Farrell did well to deny Paul Pogba and Marcus Rashford before Greenwood fired the home side ahead when he capped a slick move with a clinical finish.
Marcus Rashford beat his marker on the left flank and after Bruno Fernandes let the striker's pass roll with a clever dummy, Greenwood drilled in a side-footed shot from 10 metres past the keeper and two defenders.
United's joy was short-lived, though, as Tarkowski muscled his way into the six-yard box and rose above Harry Maguire to head home an Ashley Westwood corner, underlining United's recurring failures to deal with set pieces this season.
But Greenwood struck again in the 84th minute as his snap shot from inside the penalty area clipped Jack Cork's heel and beat Peacock-Farrel inside the near post.
Halftime substitute Cavani put the icing on the cake for United in stoppage time after another flowing move, tapping in a Donny van de Beek assist into an empty net after United broke forward from the edge of their box. (Reuters)