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.
“We needed a win to qualify, and it’s done,” Griezmann told French television.
“The coach had told me that I would be a substitute. I knew it and I wanted to do what I had to do once I got on.”
Payet also struck a late winner as France opened the tournament with a nervy 2-1 win over Romania.
Two leave-it-late wins left doubts about whether Didier Deschamps‘s side’s can repeat the feats of French sides past at the 1984 Euros and 1998 World Cup by winning on home soil.