Brady broke former New Orleans Saints quarterback Drew Brees’ record for career pass completions in the second quarter, with a 20-yard toss to wide receiver Mike Evans that took his total to 7,143.
But the individual achievement was in danger of being overshadowed by a furious Bills fightback as Buffalo erased a 21-point second half deficit to force overtime.
As he has so often, Brady came through, finding wide receiver Breshad Perriman for a 58-yard game-winning touchdown – the 700th TD pass of his career.
“It was pretty cool. I’d rather it not come down to that, but in the end they all count the same,” Brady said.
“We’ve got to learn from it and move on. Next week we’re playing for a division championship and that’s pretty exciting.”
Brady, a 22-year veteran and three-time NFL Most Valuable Player, completed 31 of 46 passes for 363 yards and two touchdowns as the Bucs extended their lead over Buffalo atop the AFC East division.
The Bucs looked headed to a comfortable win when they took a 24-3 lead into halftime.
But Bills quarterback Josh Allen ran for a third quarter touchdown and a Buccaneers field goal early in the fourth was followed by three scores by the Bills.
Allen connected with Dawson Knox and Gabriel Davis for touchdowns before Tyler Bass booted a field goal for the Bills that knotted the score at 27-27 and forced overtime.
Buffalo were unable to score on the first possession of overtime, opening the door for Brady and the Bucs, who improved to 10-3 and remain in the hunt for the top seed in the AFC.
The San Francisco 49ers also won in overtime, breaking the Bengals’ hearts in Cincinnati with a 26-23 victory.
The Bengals had opened the extra session with a field goal from Evan McPherson to take the lead, but San Francisco quarterback Jimmy Garoppolo threw a 12-yard touchdown pass to Brandon Aiyuk to clinch the victory.
Officials initially ruled that Aiyuk had come up short, but the replay gave him the touchdown and the 49ers a needed victory.
Chiefs rout Raiders
In the late game, Aaron Rodgers threw for 341 yards and four touchdowns as the Green Bay Packers beat the rival Chicago Bears 45-30.
Elsewhere, the Kansas City Chiefs offered a reminder of just how dangerous they can be as the post-season approaches, routing the Las Vegas Raiders 48-9.
The Chiefs, who struggled early in the season and were in last place in the AFC West at one point, notched their sixth straight victory and lead the division.
Patrick Mahomes threw two touchdown passes as the Chiefs built a 35-0 first half lead.
The Chiefs defence, which sparked plenty of doubts earlier this season, forced five Vegas turnovers.
“We faced a lot of adversity early in the year (that) people hadn’t seen, and people kind of threw us down and acted like we were done,” Mahomes said.
“But you’re seeing now that we have the guys to do it.”
The New Orleans Saints kept themselves in the NFC Wild Card hunt with a 30-9 victory over the New York Jets.
New Orleans quarterback Taysom Hill was playing with an injured finger on his throwing hand, but the Saints used a strong ground game to end a five-game losing streak.
The Baltimore Ravens suffered an especially painful 24-22 loss to the Cleveland Browns as quarterback Lamar Jackson exited in the first half with what coach John Harbaugh said was a right ankle sprain.
“We’ll look at it more tomorrow and see where we’re at,” Harbaugh said of the injury, which came when Jackson was tackled from behind by rookie line backer Jeremiah Owusu-Koramoah.
Down 24-9 heading into the fourth quarter, the Ravens, with backup quarterback Tyler Huntley at the helm, came up just short of a comeback victory.