Teddy Thomas's 71st-minute try looked like it would be enough to hand France the win in front of their own fans, but a stunning long-range drop goal from Johnny Sexton in the 83rd minute ensured that Ireland came away from the Stade de France with an opening victory.
The match was France's first under head coach Jacques Brunel, but the hosts struggled to produce much of their mercurial flair throughout the match and trailed 12-6 heading into the final 10 minutes.
Both sides had been limited to penalties until Thomas provided the outstanding moment of quality, collecting a kick on the halfway line before bursting past Rob Kearney and outpacing three more Ireland defenders on his way over the line.
However, it would not prove to be the decisive moment of the contest as Sexton kept his nerve with the last kick of the game, dropping into the pocket on Ireland's 40th phase before slotting his kick over from just inside the France half.
France must now attempt to bounce back when they visit Scotland next weekend, while Ireland take on Italy as they look to complete the second leg of a potential Grand Slam.