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Griezmann, who scored in last Wednesday’s opener against Ukraine, netted the world champions seven points in three games.
Ukraine, who were held to a 1-1 draw at home by Kazakhstan, have three points with Finland on two points in two matches.
Bosnia, which has not won in its last 11 outings, is in fourth place with one point.
France’s performance was far from perfect and coach Didier Deschamps was not satisfied.
“Our first half was below par. It was better after the break but I saw things that I hadn’t seen in a while. I know it’s difficult to play a third game in eight days, with all the travel, but I warned them. that this match was important, ”he said.
Captain Hugo Lloris played his 123rd game for France, joining Thierry Henry in second place on the list of Blues‘most capped players behind Lilian Thuram (142).
The keeper shone in the 24th minute with a superb save to deny Darko Todorovic as Bosnia approached the opening goal, counter-attacking after Kingsley Coman tested Ibrahim Sehic.
Lloris was up to the task again with an impressive point blank header from Anel Ahmedhodzic two minutes later.
His performance kept France afloat and Lloris was convinced the team had shown character.
“We struggled and they were dangerous on the counterattacks and on the set pieces, but the mood was perfect,” he said.
“We played badly but our character made the difference. Our state of mind was great. ”
France struggled against a team happy to sit down but broke the deadlock as Griezmann returned home after Adrien Rabiot’s perfect cross.
Griezmann is now alone in fourth place on the list of all-time best male scorers in France, ahead of David Trezeguet (34) and behind Michel Platini, third, by six goals.
Bosnia was forced to switch to a more offensive strategy but Blues calmly controlled the game, hoping Kylian Mbappe’s speed would hurt their opponents at the break.
Mbappe, however, had another disappointing night, without threatening throughout.
Thomas Lemar, instead, made an inspired display, making the most of his playing time after being named in the starting 11, less than two months before Deschamps named his squad for the European Championship, while ‘Ousmane Dembele remained on the bench.
France will resume their World Cup qualifying campaign at home against Bosnia on September 1, after the Euro.