The Bills defense were responsible for keeping the game tight and the home fans frustrated for long spells of the first half as Brady was often hurried in his decision making.
Stephen Gostkowski put the first score on the board with a 35-yard field goal in the first quarter, but Bills kicker Dan Carpenter hit back with one of his own from 52 yards midway through the second period.
Carpenter would miss from 48 yards as the Bills looked to claim the lead before the half-time interval, and the Patriots capitalised to find a touchdown on the final drive of the first half as Brady fired a pass to White, who managed to shrug off a low tackle to complete the 20-yard score.
The Bills started the second half solidly on defense, and there was disappointment for the Patriots when Gostkowski missed for the first time this season with a 54-yard field goal attempt after an inadvertent whistle from an official halted an earlier play which would have likely ended with Danny Amendola sprinting clear.
It took a moment of magic from LeSean McCoy to get the Bills level as he dodged a couple of challenges before stretching to reach the endzone with one hand.
New England were back in front before the end of the third as Amendola collected a pass and beat a number of defenders for a 41-yard gain, before White was handed the ball on the next play to find the six yards needed for the lead to be restored.
An error from Leodis McKelvin helped the Patriots make it a two-score game before the final period as he fumbled a punt return to gift the ball to the home side, with Gostkowski bouncing back from his earlier miss to convert a 35-yard field goal when Brady was unable to force another touchdown.
With the pressure on the Bills in the final 15 minutes, Tyrod Taylor found it tough to find his receivers further up the field and was forced to battle with a shoulder problem after an awkward fall.
Carpenter split the posts from 35 yards to reduce the deficit with just over four minutes remaining, but Taylor, whose discomfort was clear after every throw, was unable to deliver on his side's final drive as the Patriots remained perfect for the season to reduce the Bills to 5-5.