Brees threw for 307 yards and four touchdowns, the last a 9-yard strike to Darren Sproles that set the record with 2:51 to go.
"Honestly, I was really trying not to think about the record or anything," Brees said. "I knew we were close. A couple guys mentioned stuff to me on the sideline. I didn't want to hear it. It's like a pitcher with a no-hitter, I guess."
It was Brees' final pass of the game and it gave him 5,087 yards passing -- with one game still to play. Marino finished with 5,084 yards for the Miami Dolphins in 1984.
Minutes after Brees broke the record, Marino offered congratulations on his Twitter account.
"Great job by such a special player," Marino wrote.
As Sproles spiked the ball, Brees put his arm over his head and started walking toward midfield while the Superdome crowd went wild and his teammates chased him down.
"I just got bum-rushed by the offensive linemen so I figured at that point, OK, I guess we finally broke the record," Brees said.
Brees' four touchdowns gave him 276 for his career, moving him ahead of Joe Montana (273) and Vinny Testeverde (275) for ninth all-time. He is the first quarterback in NFL history to pass for more than 5,000 yards twice -- he had 5,069 in 2008.
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