The veteran quarterback became the NFL's all-time passing leader on Monday night, passing Peyton Manning for the No. 1 spot.
In a victory at Atlanta earlier this season, Brees broke Brett Favre's career record of 6,300 completions.
Brees still needs 42 touchdown passes to surpass Manning's record of 539, something the Saints quarterback could achieve next season if he maintains his current level of play.
The ball caught by Smith was given to Pro Football Hall of Fame President David Baker for display in Canton, Ohio. Brees removed his helmet, held out his arm to salute the crowd and hugged his wife, Brittany, and children on the sideline.
The Saints won with a blowout 43-19 against the Redskins, scoring 14 points in the third quarter and 3 points in the last quarter.
The game continued after with the Saints unsuccessfully attempted a two-point conversion - with Brees notably split out as a receiver, at least in theory, while backup quarterback Taysom Hill took the snap and tried to run into the end zone. Brees now has 1545 yards and 10 touchdowns this season. Brees is roughly 5,000 yards ahead of the next active player on the all-time passing list - No. 4 Tom Brady at 67,077.
"You've ruined that for me".
Anticipating that Brees might break the record against his team, Redskins Coach Jay Gruden told the Athletic's Tarik El-Bashir before the game, "It's a great accomplishment, man". He has also led the league in completions six times, in completion percentage five times and in passing touchdowns four times. You've got to love the guy taking the news of his record being topped in stride.
The Saints took a chance on Brees in 2006, and the payoff has been extraordinary.
The game featured other touches meant to honour Brees.