Brees completed 33 of 47 passes for 325 yards, one score and an interception in the 30-27 win at the Mercedes-Benz Superdome. He also had a rushing touchdown.
The Saints trailed 20-3 in the second quarter before they rallied for their second consecutive win.
"We came out really dialed in in the second half," Brees told reporters. "We executed in the run game, the defense came up with some big stops and we scratched and clawed our way back."
Chargers rookie Justin Herbert had a terrific performance and completed 20 of 34 passes for 264 yards and four touchdowns. Chargers wide receiver Mike Williams had five catches for 109 yards and two scores. Emmanuel Sanders had 12 catches for 122 yards for the Saints.
"It was definitely a tough loss," Herbert said. "To be so close in the last four games hurts a lot. I know we are going to turn it around. Things are going to change."
Los Angeles and New Orleans exchanged punts on the first two drives of the game. Herbert then led the Chargers on a nine-play, 80-yard touchdown drive. He ended the drive with a 17-yard touchdown pass to Keenan Allen.
Wil Lutz made a 48-yard field goal for the Saints at the end of the first quarter. Herbert then threw a four-yard touchdown pass to Williams on the next possession to give the Chargers a 13-3 edge.
Chargers free safety Nasir Adderley intercepted Brees three drives later to give Los Angeles the ball on the New Orleans 22-yard line. Herbert threw a three-yard touchdown pass to tight end Hunter Henry three plays later to push the Chargers lead to 17 points.
Brees led the Saints on an eight-play, 51-yard touchdown drive at the end of the half. He ran for a one-yard score at the end of the drive to make the score 20-10 at halftime.
Lutz made a 53-yard field goal in the third quarter before Brees threw a 41-yard touchdown pass to tight end Jared Cook to tie the score at 20-20 with 11:29 remaining.
Herbert threw a 64-yard touchdown pass to Williams with 3:50 remaining to push the Chargers lead back to seven points. The Saints tied the game on the next possession when Taysom Hill ran for a nine-yard score.
Chargers kicker Michael Badgley missed a 50-yard field goal attempt at the end of regulation, which sent the game to overtime. Lutz made a 36-yard field goal on the first possession of the bonus period. The Chargers failed to get into field goal range on their final possession.
Saints running back Alvin Kamara had 119 yards from scrimmage on 19 touches in the win. Trey Hendrickson had 1.5 sacks and Sheldon Rankins had one sack for the Saints.
Joey Bosa and Uchenna Nwosu each had sacks for the Chargers.
"It was a win," Saints coach Sean Payton said. "There are a number of things we have to clean up. We will get back to work next week and be able to address some of the things we haven't done well and get ready to start playing after the bye."
The Chargers (1-4) have a Week 6 bye before they host the Jacksonville Jaguars at 4:25 p.m. EDT on Oct. 25 at SoFi Stadium in Inglewood, Calif. The Saints (3-2) also have a bye week before they host the Carolina Panthers at 1 p.m. EDT on Oct. 25 in New Orleans.