Griezmann swung the free kick into a mass of French and Croatian players and Mandzukic diverted the ball into the net, just past goalkeeper Danijel Subasic's outstretched hand in the 18th minute.
It was the 12th own goal of the World Cup, an all-time record, and the first ever in a World Cup final.
It was the first goal in regulation in a World Cup final since 2006 — the 2010 and 2014 finals were both extra time 1-0 results.
Croatia then leveled from Ivan Perisic in the 28th minute.
Domagoj Vida got the ball at a free kick and passed it back to Perisic on the edge of the area. The winger controlled the ball with his right foot and used his left to fire it past French goalkeeper Hugo Lloris.
It's the third goal in seven games for Perisic and the first goal France has conceded since its 4-3 win over Argentina in the round of 16.