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.