The brace took Mbappe past Olivier Giroud as France's all-time leading goalscorer and moved him to within two goals of Klose's World Cup record.
Senegal were the better side for long periods of the first half, with Nicolas Jackson striking the post and Ismaila Sarr missing a clear chance before the break.
Didier Deschamps reshuffled his front line at half-time, moving Michael Olise into a more central role and shifting Ousmane Dembele wide.
The change paid off seven minutes into the second half, with Mbappe converting an Olise pass to break the deadlock.
Substitute Bradley Barcola added a second from a pinpoint Adrien Rabiot ball, before Senegal pulled one back through Ibrahim Mbaye in stoppage time.
Mbappe settled the contest moments later, striking from distance to restore the two-goal margin and seal the win.
Deschamps, in his final campaign as France coach, admitted his side had cause for concern before the turnaround.
"It's a bit of a relief. We were a little apprehensive at the start. Changing Ousmane and Michael's positions made a big difference."
"It's always important to win your opening match. That's the magic of football when you get to share those emotions. It's only the first game but we're going to enjoy it."
Mbappe pointed to the unpredictability of tournament football despite the result.
"It gives us a bit more peace of mind, although you're never really relaxed at a World Cup. We've seen it with the other teams – winning is never easy."
"It wasn't straightforward today but we know we're capable of scoring at any moment. We have to stay calm and composed, stick to what we need to do, because everything happens so quickly at a World Cup. Now our focus shifts to the match against Iraq."
Senegal goalkeeper Edouard Mendy conceded his side had fallen short of their game plan.
"We weren't able to execute our game plan. We knew we had to raise our level in the second half and be more clinical going forward."
"We only managed to do that in spells whereas against teams like this you have to be meticulous and leave nothing to chance because you're up against pure talent. The first game is behind us now, we have to learn from it and turn our attention to Norway."
France sit top of Group I after the opening round of fixtures, with Iraq up next before a meeting with Norway.