Buffalo moved to the New England 15 before Fitzpatrick threw the ball right into the hands of McCourty while trying to hit T.J. Graham.
New England is 11-0 against the Bills at Gillette since it opened in 2002 and 23-2 overall in its last 25 games against Buffalo.
The Patriots were coming off a bye that followed their best game of the season, a 45-7 win over the St. Louis Rams in London. The Bills have lost five of six.
Fitzpatrick completed 27 of 40 passes for 337 yards, the second time this season he's thrown for more than 339. The other came when he threw for 350 in a 52-28 loss to the Patriots in the fourth game. He also threw for two touchdowns Sunday, but his only interception ruined the Bills' upset chances.
Brady was 23 for 38 for 237.
The game was closer than expected considering the Patriots led the NFL in points and yards gained and the Bills were next to last in points and yards allowed.
The Patriots extended their streak to 11 straight scoring possessions against the Bills with a field goal and three touchdowns on their first four series Sunday. In the previous meeting, New England scored on its last seven series: six touchdowns followed by a field goal.
But not even that and 13 Buffalo penalties through three quarters could give the Patriots a secure lead and they entered the fourth quarter ahead 31-24.
The Patriots led 24-17 at halftime, scoring on a 43-yard field goal by Gostkowski, touchdowns runs of 1 yard by Stevan Ridley and 15 yards by Danny Woodhead, and a 2-yard pass from Brady to Rob Gronkowski, his fifth touchdown catch in three games.
Gronkowski's touchdown came two plays after Buffalo's fourth defensive pass interference penalty and third in the end zone, covering 36 yards
The Bills scored on a 42-yard field goal by Rian Lindell, a 14-yard run by Fred Jackson and a 3-yard pass from Fitzpatrick to Scott Chandler with 31 seconds left in the half.
Chandler's touchdown capped the fourth straight scoring drive of the game covering at least 72 yards in less than 3 1-2 minutes.
Woodhead scored again, his first multi-touchdown game of his career, on an 18-yard pass from Brady. But the Bills cut the lead to 31-24 on the next possession on a 1-yard run by Jackson and Lindell's extra point.
Gostkowski's 48-yard field goal gave the Patriots a 34-24 lead 50 seconds into the fourth quarter.