Greene scored three touchdowns while embattled quarterback Mark Sanchez passed for 82 yards with two touchdowns on 11-of-18 efficiency as New York (3-3) broke a two-game skid.
Sanchez, whose starting job was on the line with Tim Tebow waiting in the wings, completed more than 50 percent of his passes since the first week of the season.
The Colts (2-3) struggled under Andrew Luck, who was picked off twice en route to completing 22-of-44 passes for 280 yards.
Adam Vinatieri nailed field goals of 20, 50 and 47 yards in the loss.
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