- Karim Benzema (66')
- Kylian Mbappé (76')
The French attack stuttered until Benzema came on as a second-half substitute and Mbappe set him up for the opener.
Mbappe then scored France's second as the world champions, who had already qualified for Qatar 2022 after Saturday's 8-0 demolition of Kazakhstan, finished Group D with 18 points from eight games.
Finland, who were hoping to secure a playoff spot, ended up third on 11 points, one behind Ukraine who beat Bosnia 2-0 away to take second place.
"We wanted to finish on a high and we're happy we did it. Now we're looking forward to the World Cup. With the team we have it's natural to think about [winning] it," Benzema said.
France have now scored in their last 18 games, matching a record that dates back to 1999-2000.
Finland were the last team to manage a clean sheet against Les Bleus and it seemed they may repeat the feat, but Benzema's entrance in the 57th minute signalled the end of their hopes.
Didier Deschamps had made five changes to the team who hammered Kazakhstan, with Benzema and Kingsley Coman on the bench.
The duo's absence was clearly felt as Les Bleus struggled to create chances against a compact Finland team.
Finland started the second half with poise but the French defence stayed alert and the visitors stepped up a gear when Benzema replaced Moussa Diaby.
They were rewarded in the 66th minute with Benzema finding the back of the net with a deflected low shot after a neat one-two with Mbappe, who found the Real Madrid striker with a back heel.
Ten minutes later, Mbappe outsprinted three Finnish players on the left flank before finishing with a superb curled shot into the far corner of the net for his 24th international goal.
It was Mbappe's fifth goal in two games after he bagged four against Kazakhstan.
"There was more at stake for Finland than for us tonight but we did what we had to do," Deschamps said. "It was a very positive year for France, except for those fatal 10 minutes in the last 16 of the European championship against Switzerland."