New England trailed 24-0 at halftime but then rattled off 31 straight points to grab the lead as Tom Brady tossed three touchdown passes and set up a 1-yard scoring run by Brandon Bolden.
The two teams then combined for three scoreless possessions in overtime before New England punted with 3 minutes remaining. Denver's Wes Welker signaled for a fair catch but the ball hit Tony Carter of the Broncos on the fly and the Patriots' Nate Ebner fell on it at the Denver 20-yard line.
Three plays later, Gostkowski put one through the uprights and AFC East-leading New England (8-3) had the win.
Brady finished 34-of-50 for 344 passing yards. Two of his TD passes went to Julian Edelman, who made nine catches for 110 yards.
Manning completed 19-of-36 passes for 150 yards, a TD and interception for AFC West-leading Denver (9-2).
Knowshown Moreno ran for 224 yards with a score on 37 carries.