Diaby, whose career has been hampered by injuries and had not played for Les Bleus since June last year, netted with a cross shot from Karim Benzema's through pass in the 20th minute.
"Last season was a tough one but I don't want to look behind, I prefer to move forward," said Diaby.
Finland also had chances but proved too clumsy to make their presence in the box count as new coach Didier Deschamps claimed his first win in his second match in charge since taking over from Laurent Blanc after Euro 2012.
France, who had Hugo Lloris to thank for in the final minutes after the keeper made a lightning quick save, next play Belarus at the Stade de France on Tuesday in the second game of a group also featuring world and European champions Spain.
"We were in danger in the end but it's always the case when you fail to add a second goal," Deschamps said.
Finland sat back and waited to hit France on the break and their plan almost paid off after 14 minutes.
Kasper Hamalainen caught the visitors' defence off guard but Yanga-Mbiwa denied him with a fine tackle just as he was about to shoot.
France went ahead in the 20th minute when Benzema slalomed through the midfield before feeding Diaby with a through ball for the Arsenal midfielder to find the back of the net.
Yohan Cabaye's fierce shot was parried away by Lukas Hradecky five minutes into the second half as Deschamps's side pushed hard for a second goal.
Finland, however, also went close when Teemu Pukki was found unmarked in the box but Anthony Reveillere managed to reclaim the ball just in time.
The hosts pressed in the final 20 minutes but young French centre backs Mapou Yanga-Mbiwa and Mamadou Sakho stayed alert until the final whistle to keep Finland at bay.
"It [Yanga-Mbiwa and Sakho] was a logical choice, and they confirmed their good performance from the previous game [a 0-0 draw against Uruguay]," said Deschamps.
"It's always harder up front. We need to be more present, although Diaby brought a lot in this area. We also need to be more spontaneous."
Lloris, who just joined Tottenham Hotspur from Olympique Lyon, once again showed his class with a quick save to deny Tim Sparv's last-gasp header.
