It was the 37-year-old veteran who drew first blood, locating Eric Decker to touch down midway through the first quarter.
Manning replied with a 12-yard pass for Julius Thomas to restore the Broncos' lead, but the contest then swung dramatically in favour of the Colts.
The 24-year-old completed his hat-trick by finding Coby Fleener to make it a 12-point ball game just before the break.
A tight game unfolded in the third quarter, but Luck piled more points after rushing with the aid of some good blocking from Reggie Wayne.
Matt Prater sent a kick over from 31 yards as the Broncos attempted to rouse themselves before the start of the final quarter.
Vinatieri restored his team's 12-point advantage early in the fourth, but the comeback was well and truly on when Manning came up trumps with his hat-trick pass for Demaryius Thomas.
The home fans were beginning to get twitchy after Knowshon Moreno burst over from a yard to make it 36-30 with just over eight minutes left on the clock.
Another field goal from Vinatieri - this time from 42 yards out - meant that Broncos needed a touchdown to have any chance of winning.
Prater kicked a 47-yard field goal with 12 seconds remaining, but John Fox's men could not recover the ball from the onside kick, meaning that they are now 6-1 for their season and second best to Kansas City Chiefs (7-0) in the AFC West division.