Brees needed 305 yards in Monday's game against the AFC South-rival Falcons and had 230 yards in the first half alone.
It took until late in the fourth quarter for Brees to set the new standard, as he hit Darren Sproles for a nine-yard touchdown with 2:51 remaining, giving him 5,087 passing yards on the season.
The Saints, who held a prohibitive 45-16 lead after the score, took a timeout to honor Brees, who got a standing ovation from the Superdome crowd.
Dan Marino set the previous record of 5,084 yards in his first full season 27 years ago. Marino took the Dolphins to the Super Bowl that year, but lost to the San Francisco 49ers in his only trip to the title game.
Brees was 16 yards shy of the record in 2008, and a year later helped the Saints win the Super Bowl.
The 32-year-old Brees, in the final year of his contract, has a record 12 300- yard passing games this season. With a touchdown pass to Marques Colston in the second quarter, Brees extended his streak of consecutive games with a touchdown pass to 42.
Colts legend Johnny Unitas holds the all-time record of 47 straight games with a TD pass from 1956-60.