But Greenwood scored his second with six minutes remaining 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.
"First half we started OK, got control but we couldn't find the big openings and Burnley made it more of an end-to-end game, bit scrappy because they defended and attacked so well we couldn't get a rhythm. We played and ran in the second half, got chances and got the goal deservedly, you switched off for two seconds for the corner and that's how it is against Burnley."
On Mason Greenwood, he added: "Fantastic, two very good goals coming in off the left at the moment. I feel his best position is to be more free on the right, face the goal and he showed that today.
"We've always said this year has to be progress and improvement, we've got as many points as last season and then if we get some more we've got more than last season, that's progress. We want to finish with a trophy in the Europa League, we've got loads of games and loads to focus on."