The French pair, ranked 20 and 30 respectively in doubles, defeated reigning Wimbledon and Australian Open champions Sara Errani and Roberta Vinci 6-1, 6-2 to book a semi-final spot.
The two 21-year-olds had only played together once before, in a Fed Cup clash with Kazakhstan two years ago, but inspired their country to a dramatic comeback victory.
France had started the day 2-0 down in the tie having never before recovered from such a deficit.
And in front of them were the 2006, 2009, 2010 and 2013 Fed Cup champions who had not lost on home soil since 2008.
Mladenovic defeated Errani 6-4, 6-3 before Garcia beat Camila Giorgi 4-6, 6-0, 6-2 to set up a deciding rubber in the quarter-final tie.
But the coup de grace came in the doubles when five-time Grand Slam winners Errani and Vinci, who have won all four Grand Slams, were put to the sword.