Didier Deschamps' side finished the first half strongly and could have extended their lead, but a clear chance fell to the wrong man as Gallese saved from left-back Theo Hernandez after a fluid France move from right to left.
The race to the top of Group C continues on Thursday morning as group-favorite France takes on Peru.
With two wins from two matches in Group C, France is through to the next round with a match to spare while Peru has been eliminated.
France, one of the pre-tournament favourites, join Uruguay and hosts Russian Federation in the knockout rounds with one group game to go.
France joined Russian Federation and Uruguay in securing a place in the World Cup second round, but it took a Kylian Mbappe strike in the first half to seal another unconvincing victory for the 1998 champions. Pogba crunched into a challenge as the South Americans tried to relieve themselves of building French pressure, winning possession and then setting up Giroud for a shot at goal.
French captain Hugo Lloris is the 7th player to reach 100 caps with France - keeping a clean sheet in this match.
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Indeed, aside from a big chance for Paolo Guerrero when the match was still scoreless and a fine effort from Pedro Aquino soon after the restart, France did enough to keep their opponents at an arm's length.
After a 36-year wait to return to the World Cup, Peru's Russian adventure will come to an end with their final group game against Australia next week.
Once again, France struggled to find their rhythm, with the trio of Giroud, Antoine Griezmann and Mbappe playing brilliantly together but too often in isolation of the rest of the side.
But France's early intensity started to fade and Peru should have taken advantage, yet Guerrero shot straight at Lloris on the turn after Christian Cueva found him in the box. When his shot was half-blocked by Miguel Trauco it ballooned up over the keeper for Mbappe to finish.
Peru coach Gareca sent on Anderson Santamaria and Jefferson Farfan at the break, while Samuel Umtiti shook off a dead leg, before Aquino clipped the outside of the post with a swerving drive from 25 yards. Same goes for Nabil Fekir who came on for Ousmane Dembele.
The referee booked Edison Flores by mistake but corrected his decision after being prompted by the video assistant referee.