Frederic Michalak, who had again endured a mixed afternoon, showed nerves of steel to notch the conversion with five minutes left on the clock.
With Ireland having just one win to their name - an opening day victory over Wales - and France still without victory, both sides pushed in the final stages.
The hosts lost Luke Marshall and replacement Ian Madigan to injury while Brian O'Driscoll, possibly playing in his last Ireland match at the Aviva Stadium, hobbled off and then back on again so depleted were Declan Kidney's ranks.
Neither side could find the crucial score, however, ensuring a 13-13 draw which left Ireland with only one win in their last 14 matches with France, and Heaslip ruing what might have been.
"It probably wasn't the prettiest game to watch but I suppose the score reflected the game, they came on pretty strong in that last 20 minutes and it was a tough game," said Heaslip whose first-half try, along with eight points from fly-half Paddy Jackson, had given Ireland their lead.
"In fairness to Picamoles he took his chance well at the end and we were professional to the last but there was just nothing between the two teams."