Group A will go down to the wire after DR Congo's 2-1 win in Libya, although Tunisia need only a point at home to Libya in the final day to secure qualification after Youssef M'Sakni's hat-trick helped them to a 4-1 win at Guinea.
But it now faces a near impossible task, as Sweden thrashed Luxembourg 8-0 earlier on October 7, with forward Marcus Berg scoring four goals, to extend their advantage over Dick Advocaat's men on goal difference to 12.
It was a nervy performance from the French, who endured long spells of pressure from the Bulgarians and saw N'Golo Kante pick up a hamstring injury after just half an hour, although victory at home to Belarus on Monday will secure their place in Russian Federation.
France, who face Belarus in Paris on Tuesday, will be confident of collecting the necessary points, and while qualification is still far from a formality, Les Bleus were in relaxed mood as they geared up for the clash at the Vasil Levski Stadium.
The hosts took the lead within 10 minutes as Ola Toivonen was fouled in the box by Christopher Martins Pereira and Andreas Granqvist stepped up to slot home.
Iceland scored another goal in the 49th minute from a corner kick to secure a 3-0 lead.
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Cristiano Ronaldo came off the bench to spare Portugal's blushes in a 2-0 win over lowly Andorra, with AC Milan's Andre Silva netting a late second.
France's rich array of attacking talent wasted little time in stamping their authority on the game, combining in sublime style to open the scoring inside just three minutes. "When you have good players, it's normal to expect us to make the difference".
The French are assured of at least a playoff place, with the best eight second-placed teams going through to the two-legged playoffs.
The visitors continued to dominate and almost found a second seven minutes later when N'golo Kanté found Mbappé in space on the left and the PSG forward forced a fine save from Plamen Iliev at the near post.
Greece could also finish second in the group as they trail Bosnia-Herzegovina by a point ahead of their trip to Cyprus.
Goals from Haris Medunjanin and Edin Visca put the Bosnians ahead in Sarajevo after Belgium full-back Thomas Muenier's early strike, but Michy Batshuayi and Jan Vertonghen turned things around.
The other game in the group sees Gibraltar, who are without a point, host Estonia.
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