Didier Deschamps' men headed into the game three points adrift of Group I leaders Spain, and knew that victory would likely not be enough to see them qualify automatically.
Franck Ribery gave France the lead with a superb finish from distance after just eight minutes, before Joona Toivio's own goal and Karim Benzema's late strike ensured triumph.
France dictated for the majority of the game in what was a strong performance, but their efforts proved in vain as Spain's 2-0 win over Georgia ensured that the world and European champions topped the group.
The hosts made a quick start to proceedings, Paul Pogba firing wide as Deschamps' men looked to seize the advantage early.
France showed great attacking intent in the first half, and only the reflexes of Maenpaa prevented them from doubling their lead as he saved well to keep out Valbuena's header.
The visitors looked to gain confidence from that near miss, and they almost took the lead in spectacular fashion as Alexander Ring's improvised volley dipped just over the bar.
Still, Maenpaa was the busier of the two keepers, and demonstrated more wonderful reflexes to repel Olivier Giroud's point-blank strike after the Arsenal forward had latched on to Ribery's pull back.
Deschamps' side found plenty of space in the Finland defence, however they were once again thwarted by Maenpaa, who performed further heroics to parry Mathieu Debuchy's fierce drive.
The second goal did finally arrive after 75 minutes when Toivio turned Giroud's header into his own net after the striker had risen to meet Debuchy's fine right-wing cross.
Substitute Remy hit the post with a header, before Cabaye wasted a gilt-edged chance to score, but a third arrived three minutes from time as Benzema hooked home Ribery's left-wing cross.
