The Three Lions were knocked out of the World Cup by the reigning champions but Kane had the opportunity to take the quarter-final to extra-time in the 85th minute.
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Having already scored one penalty in the match, Kane blasted his second over the bar - and Mbappe's reaction went viral.
Some outlets suggested the France superstar was mocking the England captain as images circulated on social media.
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However, Laurens insists Mbappe was simply screaming with relief.
"He's not laughing," Laurens told talkSPORT. "He's celebrating and screaming. It just looked like he was laughing.
"Everybody was talking about him and he wanted to have a good game, he kind of influenced the first goal.
"But it was just relief more than anything else."
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Nevertheless, Laurens admits the French were gobsmacked when Kane stepped up to take the second penalty against Tottenham teammate Hugo Lloris.
He referenced a Europa League match against Eintracht Frankfurt in October, when Kane scored from the spot and then smashed a second over the bar.
Laurens said: "The French could not believe Kane was taking the second one. They were convinced someone else would take it. They said it's impossible.
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"If you remember the Eintract Frankfurt game in the Champions League, he blasted the second penalty over the bar.
"Already there is something for him about taking two penalties in the same game against the same goalkeeper.
"If that goalkeeper is your friend, and they are really good friends, they see each other outside of football, maybe there is something in that."
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However, Scotland legend Ally McCoist insists Kane missing was the lesser of two evils.
McCoist responded: "If England have the penalty to somebody else and they missed it, the criticism is double.
"They would've been crucified."