Despite going down to another early try, the men in green showed great mental strength to overpower a side which they drew with 20-20 in the Six Nations.
Half backs Tomas O'Leary and Gareth Steenson (12 points) were the key, maintaining a rugged pack drive lead by Blackrock skipper David Gannon.
Jean-Baptiste Peyras opened the scoring on four minutes, scurrying through to touch down, which full-back Anthony Floch converted.
Steenson quickly replied with a 25-metre penalty and wily Cork Con number nine O'Leary sliced through poor French tackling nearing the half hour to see the Irish deservedly 10-7 in front.
Pau flyer Peyras nipped in for what looked like a decisive second try on the cusp of half time, converted again by Floch, in what was just their fourth attack of note.
Territorial dominance saw the Irish on top in the second period though, and 15 minutes in Loughborough wing Paul McKenzie, a late call-up for the injured Tommy Bowe of Ulster, was put over by the O'Leary-Steenson combination.
Steenson added a sure-footed conversion and penalty and added the extras to Dave O'Brien's 71st minute try, worked well off a five-metre scrum.
French centre Denis Lison got in for a consolation score but it is the Irish through to a Wednesday semi-final at Hughenden.
Team manager David Haslett spoke of the side's achievement afterwards. "It's absolutely tremendous - it has to be the most memorable day in the history of the Irish U21s," he said.
"We were down 14-10 at half time but came back so well and it really shows you the measure of these guys. They played out of their skins, particularly the pack in the second half."
Ireland: Try: O'Leary, McKenzie, O'Brien; Cons: Steenson 3; Pens: Steenson 2
France: Tries: Peyras 2, Lison; Cons: Floch 2
Ireland: A Finn; R Lane, J Hearty, G Telford, P McKenzie; G Steenson, T O'Leary; J Wickham, D Fogarty, D Fitzpatrick, D Gannon (Capt), S O'Connor, B O'Connor, D O'Brien, J Heaslip. Sub used: K Doyle for Fitzpatrick (65 mins)
France: A Floch; JB Peyras, PE Garcia, D Lison, J Arias; R Lespinas, J Dupuy; A Falieres, B Kayser, G Bergos, R Roques, P Vigoroux, Y Nyanga (Capt), G Bernad, M Baget.