New England (6-3) led 31-17 in the third quarter, but the Bills scored 14 of the next 20 points to get into position for a late victory.
The Bills took over with 2:06 remaining and Fitzpatrick guided them on a drive to the Patriots' 15-yard line. But that's where it came to an end as McCourty picked off a pass in the end zone.
Tom Brady picked up 237 yards with two touchdowns on 23-of-38 efficiency as the Patriots who won their third straight overall and 11th in a row in Buffalo.
Fitzpatrick ended 27-of-40 through the air for 337 yards and two scores for the Bills (3-6), who have yet to record an AFC East divisional victory.