Brees hit Alvin Kamara for an 18-yard touchdown with 1:05 left to play and then handed the ball to the rookie for a two-point conversion that tied the score at 31.
It would be hard to complain about the Washington Redskins defense early in the game.
Meanwhile, Washington had racked up 338 yards, with Cousins throwing two touchdown passes and 224 passing yards. It was its mistakes and bad decisions late that brought on the loss after Washington led the game up until the last few minutes of the contest.
Adding injury to insult, Chris Thompson - the team leader in receiving yards and total touchdowns - was carted off the field with an ugly leg injury and did not return.
New Orleans was already going to be challenged downfield by the Redskins' Kirk Cousins, who was 4 of 7 for 83 yards and a touchdown in the first quarter.
The 10-yard penalty and 10-second runoff killed the drive.
The Redskins gave up 15 points in the final three minutes of regulation on Sunday to set up a dramatic comeback for New Orleans. The drive that followed was capped by a 40-yard Cousins TD pass to Ryan Grant, who was all alone behind the coverage.
Aggressively looking down field against a hobbled Saints secondary, Cousins completed 22 of 32 passes with no interceptions.
Redskins cornerback Josh Norman (24) walks off the field after the team's overtime loss in New Orleans.
Fans were heading for the exits as the Redskins took a 15-point lead with 5:58 remaining after quarterback Kirk Cousins delivered a lovely, six-play, 75-yard drive that ended on a seven-yard touchdown pass to rookie tight end Jeremy Sprinkle.
Now 31-16 after the PAT, there was 5:58 left in the game. "It wasn't quite close enough for a field goal, so I was trying to get (wide receiver Jamison Crowder's) attention and hand-signal a bubble screen out there and get out of bounds and get another play called". Even with a big game next week looming against the Los Angeles Rams, the Saints have proven to this point that they're a legitimate threat to go the distance.
The Redskins gained 456 yards - their second most this season - against a Saints defense that had not allowed more 347 yards in any game during New Orleans' winning streak.
It won't get any easier for Washington, which must finish its season without its top running back. And the defense crushed the ensuing Redskins drive to send the game to overtime. They were aggressive and moving the ball at will.
Two long Mark Ingram runs put the Saints in field goal range, and Will Lutz kicked the game-winner from 28 yards out.
How else would one explain a bank of lights suddenly going dark inside the Superdome with 2:53 remaining in regulation after Hill's 3-yard TD catch pulled the Saints within 31-23. "No disrespect. It's something we've got to cope with and try to fix".
"We had 25 seconds left and first down and we were in position where it was the 35 (yard line)", Washington head coach Jay Gruden said.