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 had been cruising at the top of their World Cup qualifying group but conceded two goals in the final seven minutes to lose 3-2 at home to Israel, meaning they had to avoid defeat in the final game, at home to Bulgaria.More news: Galen Rupp becomes first American man to win Chicago Marathon since 2002
David Propper gave the Netherlands a 1-0 lead in the 25the minute, a lead that was tied up early in the second half when Maksin Volodko drove the ball past the Netherlands' goalkeeper.
Les Bleus took an early lead through Blaise Matuidi and controlled the pace the rest of the way, as the match turned into a drab affair.
The opening goal came in the 24th minute when Ryan Babel played a cross to the far post for Arjen Robben, who with one touch centred for Davy Proepper to score.
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.
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. It's true that it was a fight, and we need to do better.More news: Canada Cancels Plans to Build Pipelines
France top the standings, but still have a long way to go, as the top four teams in the group have a shot at qualification.
The late game in that Group saw Greece come from a goal down to beat Cyprus 2-1 and go second in the group with a game to go.
Cyprus, on 10 points, could have stayed in contention but an bad mistake by Konstantinos Laifis six minutes later let Greece back into the game.
In the other Group H afternoon kick-off, defender Joonas Tamm bagged a second-half hat-trick as Estonia trashed Gibraltar 6-0 at Estadio Algarve.
Portugal is now second in their qualifying group behind leaders Switzerland.More news: Former Equifax chief apologizes for data breach to House panel