The Italian came through by a 6-3 5-7 6-4 scoreline in a contest that lasted almost two-and-a-half hours.
Over the weekend, Vinci benefited from the withdrawal of Eugenie Bouchard, who suffered concussion ahead of their scheduled last-16 contest, and the Italian looked the sharper of the two players during the early stages.
The 32-year-old, who was making her third appearance in the quarter-finals at Flushing Meadows, broke in Mladenovic's opening service game and it proved enough for her to take the first set.
At the start of the second, Mladenovic created six break points, but she wasn't able to convert any of them before she lost her own serve in the third game.
However, three games later, she was a break to the good after starting to dominate the Vinci serve, and although she was broken back and had to receive a medical timeout, she was able to strike again in the 12th game to take the match to a decider.
During the opening seven games of the third, eight break-point chances were created, but it was Vinci who recorded the one and only break to move to the brink of victory.
One match point went begging as Mladenovic threatened a fightback, but Vinci secured the triumph on the second to set up a semi-final showdown with either Serena or Venus Williams.