Antoine Griezmann (58th, 61st minutes) struck in quick succession in the second half of the entertaining second round clash after Robbie Brady (2nd) converted from the penalty spot to hand Ireland an early lead. Brady's goal was Ireland's second against France since October 1981.
France are now unbeaten in their last 16 games at major tournaments on home soil. They have won 14 of those while two matches have ended in draws.
The 'Les Bleus' will face either England or Iceland in the quarter-finals, but will miss the services of Adil Rami and N'Golo Kante, both of whom will be suspended.
The hosts have not exactly been outstanding in the group matches. But the nightmare start forced them to go on the offensive and they produced their best performance so far in the tournament.
But France coach Didier Deschamps still has issues to worry about as both goals France have conceded at this year's tournament were from clumsily-conceded penalties. Bogdan Stancu scored for Romania against Les Bleus in the tournament opener but France still managed to win.
Ireland had ended their campaign in the group stage with a 1-0 victory over strong title contenders Italy and they set out with the objective of grabbing another unlikely upset.
This was the first meeting between the sides since Thierry Henry's handball during a World Cup qualifying match in 2009 sent France on their way to the 2010 World Cup at the expense of the Irish.
The sizeable contingent of Irish fans in the stands certainly had reasons to believe that revenge was on the cards when Shane Long was brought down by a clumsy challenge from Paul Pogba and Brady, who plays for English Championship side Norwich City, stepped forward to dispatch the resultant penalty into the lower right corner.
The early setback forced the hosts to go on the offensive in search of the equaliser. They enjoyed better possession throughout the game, but some sturdy defending by Ireland kept them at bay in the first half.
Deschamps made an attacking change at the break, replacing N'Golo Kante with 20-year-old sensation Kingsley Coman. That proved to be decisive as the recent Bayen Munich recruit provided an extra cutting edge to the French forwardline.
The equaliser seemed almost imminent as the French continued to chip away at the Ireland defence. An attempt from distance by Pogba flew just wide of the post, another by Blaise Matuidi was palmed away by Irish goalkeeper Darren Randolph.
Griezmann finally broke the desparate Irish resistance with a couple of superb strikes in a space of three minutes.
Bacary Sagna made a powerful run down the right before whipping in a cross which was met with a excellent header by Griezmann.
Just three minutes later, Olivier Giroud used his strength to head down Rami's cross from the deep to set up Griezmann, who surged into the area, dribbled past Randolph and found the bottom-right corner with a precise left-footed finish.
The Irish never recovered from the double blow.
With France dominating possession, the action was mostly confined inside the Ireland half. The Irish were facing an uphill battle and their task was made even more difficult in the 66th minute when Shane Duffy was shown the red card for a clumsy, off the ball challenge on Griezmann.
France almost added a fourth in the 76th minute. Coman set André-Pierre Gignac clear inside the Irish penalty area with a defence splitting pass, but Randolph somehow managed to avert the danger. But the ball was not cleared away properly and Matuidi hit the bar with a rasping shot from the edge of the box.
Soon after, a sliding Griezmann missed out a hattrick by mere inches when he could not a curving cross into the six-yard box by Gignac with Randolph virtually at his mercy.
Randolph produced an excellent save to deny Griezmann in the final minute of regulation time. But the Frenchman would certainly not mind that as France kept on track to capture their second major trophy on home soil since the 1998 World Cup.