After seeing a three-point lead cancelled out by Stephen Gostkowski's field goal at the end of the fourth quarter, the Broncos produced another response to clinch the win when Anderson sprinted clear for the game-winning touchdown.
It was a disastrous start to the evening for the Broncos, who lost TJ Ward, Sylvester Williams and Louis Vasquez to injuries picked up in the first half, and they were on the back foot early on when the Patriots got the first score on the board.
New England extended the lead to 14-0 in the second quarter when Brady picked out Scott Chandler moments after Chandler Jones had won the ball for his side when Brock Osweiler, who started in the absence of the injured Peyton Manning once again, was hurried into a mistake.
A solid start on defense for the Patriots came to an abrupt halt when Denver reduced the deficit thanks to Ronnie Hillman, who skipped past a couple of tackles on a 19-yard sprint to the endzone.
With snow beginning to fall more heavily after the break, both teams struggled to find fluency on offense and Denver kicker Brandon McManus sent a 47-yard field goal wide midway through a scoreless third quarter.
Brady delivered the goods at a vital stage with an inch-perfect pass out wide to Brandon Bolden, who completed the catch close to the sideline before holding off two tackles to sprint clear for the 63-yard touchdown at the start of the final period.
The Broncos showed their resilience to hit back once again as Shaquil Barrett recovered the ball after Chris Harper had fumbled a punt return, and the home side capitalised as CJ Anderson evaded three tackles on the way to diving into the endzone.
There was concern late on when Gronkowski was forced off the field with a knee injury and he remained on the treatment table as Osweiler connected with Emmanuel Sanders to set up first and goal.
A penalty on the Patriots defense handed Osweiler a second attempt at first down, and he capitalised by finding Andre Caldwell in the endzone to give Denver a three-point advantage with a minute and nine seconds remaining.
Brady moved his side into position for the field goal in a dramatic finish, and Gostkowski held his nerve to split the posts from 47 yards with time expiring.
Denver recovered from the setback to put the pressure on Brady in the first drive of overtime, and they took full advantage after earning possession of the ball to secure the win when, in a repeat of the fourth-quarter play, Anderson pulled away from the defenders to delight the home fans with the game-winning score.