The hosts soon built from the opening touchdown by grabbing another with just over seven minutes until the break, this time as Brady picked out a telling 11-yard pass for Danny Amendola, who made no mistake in getting over the line.
It looked as though New England might go on to build a further lead before the interval, but the Eagles hit back with a score of their own before the half-time whistle, as Zach Ertz snatched a five-yarder from Sam Bradford to make it 14-7.
The visitors' comeback was complete with just eight seconds remaining on the clock, and the equaliser came in extraordinary fashion when Najee Goode recovered a blocked kick to bring the scores level.
After the break, the Eagles were allowed take full control of the match as Malcolm Jenkins's 99-yard interception run opened up a seven-point lead, before another touchdown from Darren Sproles saw the visitors move 28-14 ahead.
Having already recovered from a similar deficit, Philadelphia ensured that the hosts would not match their comeback as Jordan Matthews collected a Sam Bradford pass to make it 35-14.
With five minutes to go, there was still time for a late rally. First, Scott Chandler latched on to another Brady ball, while the quarterback himself made it 35-28 with a one-yard rush but, with three minutes left, time was against the hosts as the Eagles held on.