Atletico Madrid forward Griezmann came off the bench in Marseille to head the opener in the 90th minute before West Ham United's Payet strengthened his reputation by smashing home a second deep into stoppage time.
Payet also struck a late winner as France opened the tournament with a nervy 2-1 win over Romania.
"This is becoming a bit of a habit, I'd prefer us to break the deadlock a bit earlier," said a relieved French coach Didier Deschamps.
However, two leave-it-late wins left doubts about whether Deschamps's side can repeat the feats of French sides past at the 1984 Euros and 1998 World Cup by winning on home soil.
Deschamps surprisingly left Juventus star Paul Pogba and Griezmann on the bench in the only two changes from the game against Romania.
Les Bleus only kicked into gear with Pogba and Griezmann reunited with the mercurial Payet after the break.
"We needed a win to qualify, and it's done," Griezmann told French television.
Griezmann finally made the breakthrough as the clock ticked into stoppage time, stooping to turn home Adil Rami's cross. And Payet capped another man-of-the-match performance by firing into the far corner to round off a harsh scoreline on Albania.
Bogdan Stancu's penalty gave Romania the advantage at the
Parc des Princes. The Swiss spurned a host of chances to score before Admir Mehmedi finally slammed home to level just before the hour mark.
Stancu is fast becoming a penalty specialist at these championships, having also scored from the spot in the 2-1 defeat to France to join Payet as the only player to score twice at the tournament so far.
But the Swiss burst back into the game when Macedonian-born Mehmedi slammed home a loose ball from a corner in the 57th minute after the Romanian defence failed to clear. Russia's troubles deepened as a day after the 2018 World Cup hosts were threatened with expulsion from the competition for crowd trouble at their Group B opener against England, they moved to the brink of bowing out for their efforts on the field with a 2-1 defeat to Slovakia.
Russia pulled a goal back through Denis Glushakov late on, but a flare briefly lit after they scored could yet provoke further punishment from UEFA.
The result allowed Jan Kozak's team to bounce back from an opening 2-1 defeat to Wales and leaves Russia languishing at the bottom of the group with just one point from two games.
Wales can seal top spot in the pool with victory over England on Thursday.
"I am very proud that I managed to score such a goal. It was brilliant.
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