Greenwood's 84th-minute strike was enough to clinch all three points for Michael Carrick's men at the Riverside Stadium on Saturday afternoon as Leicester City suffered back-to-back league defeats ahead of the international break.
The Foxes arrived on Teesside licking their wounds from a Friday night loss to Leeds last weekend, but were undone by a spectacular Greenwood free-kick as the game entered the final stages.
Leeds' on-loan midfielder joined Boro during the last few days of the summer transfer window and has scored a handful of goals already at the Riverside, cementing his place in Carrick's side. Boro also retain a £1.5 million option to buy in the attacking midfielder's loan agreement.
Greenwood's strike means Leicester's lead over United at the top of the table has been cut from 14 points prior to Leeds' visit to the King Power Stadium, to just eight as the teams break for November's international fixtures. Second-place Ipswich Town are now level with the Foxes on 39 points after 16 matches.
Leeds, meanwhile, defeated Plymouth Argyle 2-1 at Elland Road on Saturday afternoon with goals from Dan James and Joel Piroe securing all three points. The Whites remain unbeaten at Elland Road this season and have won each of their last five on home turf.