Brady finished 29 for 35 for 376 yards. Tight end Rob Gronkowski had seven receptions for a career-high 162 yards and a touchdown.
"I think I know how to get to that place where I need to be," Brady said. "You only have so much energy, and at some point you gotta cut it loose when the moment's right."
Cincinnati dropped to 2-4 for the first time since 2010, when it finished 4-12 - its worst season under coach Marvin Lewis.
But New England (5-1) bounced back from some sluggish drives early with several big plays in the third quarter.
Cincinnati has lost in its past six trips to New England. It hasn't won in Foxborough since 1986.
Copyright 2016 by The Associated Press