Ireland looked comfortable for an hour - a Jamie Heaslip try and eight points from Paddy Jackson giving them a deserved lead - but a late try by Louis Picamoles converted by a largely misfiring Frederic Michalak drew France level.
It was the second successive draw between the two sides but extended France's winless streak over the past two Six Nations to seven, their worst run since 1926-27, and leaves them needing to beat Scotland at home next Saturday to have any chance of avoiding the wooden spoon.
The draw will ease some of the pressure on Ireland coach Declan Kidney.
However, captain Heaslip said it was an opportunity missed for what would have been only their second win in their past 14 meetings with the French.
"Yes, you would say we are disappointed, feels like a chance gone abegging," said Heaslip.
"I wouldn't say we let it slip but we didn't take our opportunities in the second-half. It's a bit of a flat dressingroom at the moment."
France's try-scorer said the visitors' comeback proved they were a force to be reckoned with.
"We had a good reaction after half-time. It's a shame we didn't manage the game well in the first period," Picamoles told French television.
"But we showed our worth and we went for it. The second half showed we're a true team. We'll keep going and try to manage things better but we're happy because we proved we're a real team."
The Irish made the early running in terrible weather conditions and were rewarded after a superb tactical kick deep into the French 22 by Brian O'Driscoll set up a lineout.
Peter O'Mahony won the ball and from the resulting driving maul Heaslip squeezed over the line to score his eighth international try - Jackson belied concerns over his placekicking, after a disastrous performance against Scotland, by landing a superb conversion.
France were presented with a golden opportunity to reduce the deficit in the 16th minute but Michalak, whose selection had raised eyebrows after three poor performances, missed his penalty from close range.
Ireland were very much in control but Jackson missed a chance to make it 10-0 in the 23rd minute as his penalty from long range drifted wide.
Michalak made no mistake with his second chance in the 27th minute as he slotted over to make it 7-3.
However, the Irish scrum were forcing the French into repeated infringements and Jackson, assuming responsibility when the Irish could have kicked for the corner, converted a long range kick at goal to restore the seven point advantage on 30 minutes.
The 21-year-old, playing in only his second test, was showing little fear and he punished the French again three minutes later as he stroked a penalty over from about 45 metres for 13-3 which even had injured first choice fly-half Jonathan Sexton grinning and applauding from the stands.
Michalak's woeful first-half finished on a suitably low note as given the chance to give France a filip on the stroke of half-time he sent another kickable penalty wide to leave the Irish sitting comfortably enough at the break.
Jackson had an early chance in the second-half to extend the lead but his effort fell just short.
The Irish were still controlling the game but prop Mike Ross conceded a penalty in the 54th minute and Morgan Parra stepped up, replacing the misfiring Michalak, and converted it for 13-6.
Parra, though, proved he too was fallible, missing a tough penalty attempt just before the hour mark as the French started to pressure the hosts into a series of errors.
The visitors pressed forward despeate to redress the balance but saw a chance go begging as with extra men wide out to the right they failed to push home the advantage and the Irish defence held firm.
However, eventually the French breached the line as Picamoles took advantage of Ireland dithering after a penalty was awarded and tapping it quickly he touched down for his fifth try for his country. Michalak stepped up to the task and converted for 13-13.