Turkey started the game strong, scoring their first goal in the 30th minute with a header from Kaan Ayhan following Cengiz Under's free kick into the penalty area.
Under then raised the difference in the 40th minute, slipping the ball past Hugo Lloris after a sprint down the right wing to the French goal.
Les Bleus upped the pressure in the second half of the game, but missed several chances to score due to strong defence from their opponents and a solid performance from goalkeeper Mert Gunok.
Burak Yilmaz missed the chance to put Turkey further ahead at the 85th minute, sending the ball narrowly past the right post after a one-on-one with Lloris.
In other games in the group on Saturday, Iceland earned a 1-0 home win against Albania, while Moldova beat Andorra 1-0 at home.
Turkey now lead Group H with nine points having won all three games so far. They are followed by France then Iceland, both at six points.
Germany eased past hosts Belarus 2-0 with goals from Leroy Sane and Marco Reus on Saturday to make it two wins from their first two Euro 2020 Group C qualifiers.
Earlier on the same day, Northern Ireland came from a goal down to beat Estonia 2-1 in Tallin.
Belgium eased to a 3-0 win over Kazakhstan in Brussels to keep up their 100 percent start to the Euro 2020 qualifiers and top the Group I standings.
Croatia and Hungary pulled away at the top of Euro 2020 qualifying Group E after victories in what is shaping up to be a fiercely competitive pool.
Both teams have six points from three games after a goal and an assist by Ivan Perisic gave 2018 World Cup runners-up Croatia a 2-1 home win over Wales while a Willi Orban brace helped the Hungarians to a 3-1 win in Azerbaijan.
Wales and Slovakia have three points each from two games and Azerbaijan none.
