Ireland, who last year beat all three southern hemisphere giants but suffered a surprise defeat in Scotland in their opening game of the championship, went ahead after 29 minutes through a converted try from scrum half Conor Murray.
Flyhalf Johnny Sexton, returning to the side after a spate of injuries, gave Ireland a 10-point lead with two penalties and a drop goal early in the second half.
The loss left France, on five points, in the bottom half of the table where they have finished every Six Nations championship for the past five years.
Ireland top of the standings with 10 points but England, on eight, will move ahead if they secure their third win from three against bottom-placed Italy on Sunday.
Scotland, who beat Wales 29-13 earlier on Saturday, are in second place on nine points.
"They are a really good side and we were really happy to grind out that win,” Ireland captain Rory Best told reporters.
“The two sides were under pressure and we were very happy to come out on top."
France punished a scrappy opening from Ireland with two penalties from flyhalf Camille Lopez in the first 20 minutes. Centre Remi Lamerat briefly silenced the crowd before his try was disallowed due to a knock-on.
But momentum swung back to Ireland after 29 minutes when Murray pounced over the line from a well-worked scrum and Sexton converted.
Sexton, seeking to bolster his case for a second British and Irish Lions tour this year, sent the crowd into raptures with long-range drop goal after 49 minutes, with national radio station Newstalk tweeting "He's Baaaack!!"
Ireland held off intense pressure from the French in the last 20 minutes before conceding three points to a Lopez penalty in the 73rd minute. Sexton's replacement Paddy Jackson then restored the lead to 10 points four minutes from time.
Ireland coach Joe Schmidt said Wales were unlucky to lose the earlier match against Scotland and he believes his team's next game in Cardiff in two weeks will be a really tough one.
"We've now doubt that they will be disappointed with that and will be very determined when we get to Cardiff in 13 days time," he said.
(Editing by Ed Osmond)
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