"We were disappointed with that. We did a lot of good things but didn't manage to close the deal," said Ireland head coach Declan Kidney after his side threw away a 13-3 half-time lead.
"I think we're getting stronger in our belief each week. The frustration is obviously mounting but I don't think we're too far off a very good performance."
Ireland had looked on course to bounce back from successive defeats to Scotland and England after establishing a decent lead at the break but France dug deep in the second half to ensure they will at least avoid a tournament whitewash regardless of what happens when they face the Scots in Paris next week.
Captain Jamie Heaslip gave his side a brilliant start when he burst over from close-range after just 10 minutes following a superb driving maul from the Irish pack as they adapted best to he atrocious weather conditions.
Pouring rain meant running rugby was difficult throughout and the rest of the first-half points came via a conversion and a penalty from Paddy Jackson and a single three-pointer from France fly-half Freddie Michalak.
But Ireland failed to push on after the interval, with a Morgan Parra penalty bringing France back to within a single score after 52 minutes.
With the weather refusing to relent, Ireland were favourites to hang on for a second win of the Championship but French perseverance paid off when No8 Picamoles beat Keith Earls to a kick ahead to make it a two-point ball game.
That score handed Michalak the chance to make amends for an otherwise disappointing kicking display and the mercurial Toulon halfback did just that with a wonderful conversion from just inside the left touchline.
Ireland suffered further woe in the final minute of the match as replacement scrum-half Eoin Reddan suffered a broken leg that will keep him out of action for three months and ends his outside hopes of a Lions spot.