Griffin -- taken second overall in April's NFL draft -- went 19-of-26, including an 88-yard scoring pass to Pierre Garcon in the first quarter.
The rookie quarterback also added 42 yards on 10 carries for Washington (1-0).
Garcon made four grabs in all for 109 yards before leaving with a right foot injury.
Drew Brees completed 24-of-52 passes for 339 yards with three touchdowns and two interceptions for the Saints (0-1), who were playing their first regular-season since the team was hit by allegations of setting bounties to injure opponents.
Lance Moore had six catches for 120 yards for New Orleans.
Wisconsin business offering 'therapeutic cuddling' forced to close
Trader Joe's: Car crashes into Long Island store, injuring 11