There was little to separate the two players in the first two sets, despite Baptiste sitting at World No.283 in the rankings – over 200 places below her French foe.
After dropping serve in the fifth game of the opener, Baptiste rallied to draw level at 4-4 before ultimately falling behind by a set, but finally delivered in the clutch in the late stages of the second set.
After missing out on her first six break point chances in the second set – three in the second game, one in the eighth and her first two set points – Baptiste finally converted her seventh overall opportunity in the set, and broke Mladenovic in the 10th game to extend the match.
Ultimately, though, the 26-year-old’s experience saw her through to the last eight in the third set, as she lost just six points in four service games and never faced a break point.
The World No.49 scored the decisive break of the decider early on, as she broke the teenager from a 30-0 deficit in the third game, and was hardly challenged on serve from that point on.
The Frenchwoman, who has not reached a semifinal in 2019, next faces Russian qualifier Anna Kalinskaya, who upset No.8 seed Monica Puig in over two-and-a-half hours earlier on Wednesday, 2-6, 7-6(5), 6-4.
With unseeded Americans Christina McHale and Jessica Pegula dueling in the other quarterfinal, an unseeded finalist is guaranteed out of the bottom half of the draw.