Barcelona forward Griezmann equalised 24 minutes from time in Budapest to cancel out Attila Fiola's first-half opener and salvage a point for Les Blues in the Group F match.
Hungary was spurred on by a raucous crowd, with around 61,000 spectators in attendance thanks to an easing of coronavirus restrictions in the Hungarian capital.
France had to settle for a point that leaves it waiting on a place in the Round of 16, and Griezmann puts his side's disjointed display down to a couple of factors.
"It was a difficult match for us," he told beIN SPORTS. "With the fans here in a full stadium, we lost our normal habits. We struggled to get along and it was also super-hot.
"We got caught out at the back and didn't know how to find each other. We didn't know how to put away our chances in the first half.
"We know that we have players on the bench to make the difference at any time. We have to continue to work hard and improve."
Griezmann's goal was his seventh in European championships, putting him behind only Cristiano Ronaldo (11) and Michel Platini (nine) in the list of the competition's all-time top scorers.
That strike did at least rescue France a point that moved it to the top of Group F ahead of Germany's clash with Portugal – Les Blues' next opponent – in Munich.
Germany won easily 4-2 to blow the group up, leaving France still on top but almost certainly needing a win against Cristiano Ronaldo and his Portugal side to progress, assuming Germany doesn't have the problems against Hungary that France experienced.
France was never quite at its best in the group stage before going on to win the 2018 FIFA World Cup, and Griezmann is confident that his side will improve in a similar way as UEFA Euro 2020 progresses.
"It reminds me a little of the [FIFA] World Cup when we were made to suffer in the group stage," he said. "We will have to quickly recover and go again against Portugal.
"We always knew it would be complicated in the heat and noise in the stands. I have full confidence in the group."
Fiola's goal was the first France had conceded in 527 minutes, but it battled back to make it nine matches without defeat at major tournaments.
Skipper Hugo Lloris is staying positive after the draw with Hungary, which ends a run of five successive wins in all competitions.
"We gave 100 per cent to get the draw today, even if that is not the result we wanted," he said. "We have to be content with the scoreline.
"The opposition made the most of their chance and were well organised, helped by their supporters. We're disappointed but you have to stay positive."
Asked if the heat played a part as well as the deafening home crowd, Lloris said: "That plays a part, but conceding a goal just before half-time also didn't help us.
"If we had taken the lead it would have changed the dynamics. But we came back into the game and have to be satisfied and recover for the next match."