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Following an incredible last-16 3-3 draw with neighbours Switzerland in Bucharest on Monday night, all nine penalties were dispatched confidently in the subsequent shootout, with Mbappe needing to score at 5-4 to keep Les Bleus alive in the tie.
However, his rather tame effort was saved comfortably by jubilant Switzerland goalkeeper Yann Sommer.
Watch: Euro 2021 - Switzerland knock out France in tense penalty shootout
Mbappe - who gave a somewhat subdued performance throughout a thrilling 120 minutes in Romania - said France were “incredibly sad” to be going home early as he apologised for not converting his crucial spot-kick.
"I'm sorry about this penalty. I wanted to help the team but I failed," Mbappe wrote on Instagram late on Monday night.
"It's going to be hard to sleep after this, but unfortunately these are the ups and downs of this sport that I love so much.
"The most important (thing) will be to get back up again even stronger for future commitments. Congratulations and good luck to the Swiss team."
France boss Didier Deschamps - who dismissed questions over his own future after a chastening loss - defended Mbappe, commending the Paris Saint-Germain superstar for having the bravery to step up in the shootout in the first place and insisting that no one was angry with him.
"Of course Kylian, even if he didn't score a goal (in Euro 2020), he was decisive in many actions that we had, then he took the responsibility to take this penalty and no-one is really angry with him," Deschamps said.
"I talked to the players, we know the strength of this squad, of this team, we had many magnificent moments together today.
"It really hurts, there's a lot of sadness and, well, there were many things that we did really well in this match, but not everything and if we think too much about this match it won't help too much.
"This Euro was really difficult, it's not an excuse, and now there's the last European champion and the last world champion who have to go home, it hurts but we have to accept it."
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