The Paris Saint-Germain superstar, and France captain, had his initial kick saved early in the second half but was allowed a second attempt because of encroachment into the area by a Greek defender.
He made no mistake second time around and France could have gone on to win by a greater margin after Greece defender Konstantinos Mavropanos was sent off for barging Randal Kolo Muani to the ground, denying a clear goal-scoring opportunity.
France's win in their last outing this season follows their 3-0 defeat of minnows Gibraltar last Friday and leaves them clear on top of qualifying Group B with 12 points from four games.
They have not conceded a goal so far and are six points clear of Greece, albeit having played a game more.
"It has been a very long season for the players. They have been waiting for their holidays but we did what we needed to do to win the match," coach Didier Deschamps told broadcaster TF1.
"This is a team with less experience than six months ago, but there is plenty of quality."
Having already opened their campaign with wins against the Netherlands and the Republic of Ireland in March, last year's World Cup runners-up appear certainties to qualify for next year's finals in Germany.
Their next match is against Ireland in September, by which time it will be clear where Mbappe will be playing his club football next season.
The 24-year-old, who took over as skipper of Les Bleus when Hugo Lloris quit international football following the World Cup, last week confirmed he would not extend his contract at PSG, which expires next year.
That has raised the prospect of the Qatar-backed club selling him in this transfer window, even if he has said he intends to stay with the French champions next season.
Mbappe put the talk of his future to one side to score a penalty against Gibraltar and he was the matchwinner here in what was not a vintage French display.
"This is the hardest match we have had to play since I came here," admitted Poyet, who became Greece coach in February last year.
"The difference between France and us is incredible. I don't like to lose but we learned a lot today."
