Iceland, who previously drew 1-1 with Belgium and Italy, maintained their quarter-final chances until their last group game, but Belgium's 1-0 win over the Italians secured them second spot in Group D on four points.
It took 43 seconds for forward Malard to put already-qualified France ahead with a low shot from the edge of the box, making it the quickest goal of this year's tournament so far.
But Iceland, who grew into the game as the first half wore on, continued to give absolutely everything in the second period, as Agla Maria Albertsdottir (22) and captain Sara Bjork Gunnarsdottir (31) came close to scoring.
Corinne Diacre's France, one of the tournament favourites, saw two goals ruled out by VAR towards the end of the game before Iceland grabbed a consolation goal in the 12th minute of stoppage time through a penalty by Dagny Brynjarsdottir (30).
France, who made a dream start to the tournament by thrashing Italy 5-1 and secured their place in the quarter-finals after their 2-1 win over Belgium, will face the Netherlands on Saturday.
Group D runners-up Belgium will play Sweden on Friday.
