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 scrumhalf 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 long-range 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.