Despite dominating for long periods in sweltering heat in Budapest, the French were stunned as Attila Fiola fired Hungary into a shock lead late in the first half at a raucous Puskas Arena.
Didier Deschamps’ side responded to level through Griezmann after 66 minutes, but they were unable to force a winner and had to settle for a 1-1 draw.
It was a result Hungary, who fought hard against Portugal in their Group F opener before conceding three times late on, thoroughly deserved for a determined display. It should also give them confidence going into their next game against Germany as they seek an unlikely place in the last 16.
France, boasting their feared front trio of Kylian Mbappe, Karim Benzema and Griezmann, controlled a lot of the play, however.
Their first moment of quality came in the 14th minute when Mbappe teed up Benzema outside the area and his strike was parried by Peter Gulacsi. The rebound should have been snaffled by Griezmann, but the striker fired straight at the former Liverpool goalkeeper.
Everton left-back Lucas Digne showed why he had been drafted into the side by picking out Mbappe with a superb cross, but the Paris St Germain star’s header drifted narrowly wide of a post.
Hungary then suffered a blow as captain and striker Adam Szalai was forced off through injury after lengthy treatment. The game then lost some momentum as this stoppage was followed by a drinks break planned due to the heat.
France eventually got back into their stride and went close again when Mbappe flicked Griezmann’s lofted pass into Benzema’s path but he blasted wastefully wide.
It was much against the run of play when Hungary caught France out to take a surprise lead in first-half injury time.
Fiola pushed forward to collect an Adam Nagy pass and then continued his run after playing a one-two with Roland Sallai. The 31-year-old proved too quick for Benjamin Pavard and beat Raphael Varane into the box to stroke past Hugo Lloris.
France returned for the second half with renewed vigour. Paul Pogba set a long-range effort narrowly wide and Ousmane Dembele made an immediate impression from the bench by striking a shot against the outside of the near post.
Hungary eventually succumbed as Lloris launched a quick counter-attack with a long ball upfield. Mbappe raced after it and pulled the ball across the box. Willi Orban attempted to clear but Griezmann was quick to pounce and thump home the equaliser.
Hungary were not done and Sallai was unfortunate with a strike that was deflected and easily saved by Lloris, while Mbappe went close again for France with a shot that tested Gulacsi.
France hoped for a penalty in injury time after Presnel Kimpembe went down in the area, but nothing was given after a VAR check.