Four days after being held 1-1 by Ukraine at the Stade de France, the French built their win in the first half with Sergiy Maliy's own goal adding to Dembele's opener before Kylian Mbappe missed a second-half penalty.
France moved top of the table with four points, ahead of Bosnia, who they face away on Wednesday, Ukraine and Finland.
Ukraine host Finland later on Sunday. Kazakhstan have zero points after playing their first game of the campaign.
"The job is done. We could have scored more goals if we had been more clinical after the break," coach Didier Deschamps said.
France midfielder Paul Pogba said Kazakhstan proved to be worthy opponents on an artificial pitch in a covered stadium.
"Kazakhstan never gave up. Everybody thought they'd let five goals in but that was not the case at all," he said.
He beat the previous mark of Patrick Vieira, who started 44 games in a row between 1999 and 2002.
It was Griezmann's club team mate Dembele who broke the deadlock on 20 minutes with a right-footed shot after being set up by Anthony Martial following a defence-splitting pass from Pogba.
Dembele returned to the France side as a second-half substitute against Ukraine, more than two years after his previous cap in November, 2018.
His third international goal from 23 appearances, his first since a 3-1 friendly win against Italy in June, 2018, came as a welcome boost less than two months before coach Deschamps names his squad for this year's European Championship.
"We know that Ousmane can bring a lot to this team, he still needs to improve, we know it," Pogba said. "I'm taking him under my wing to boost him."
Les Bleus doubled the lead on the stroke of halftime when Maliy, who had just denied Martial, headed a cross into his own net.
France were never threatened and had a big opportunity to add a third in the 76th minute, but Mbappe's spot-kick was saved by Aleksandr Mokin.