Tsonga saved two match points in the 10th game before holding for 5-5, and another in the 12th game to force a tiebreak that he won 7-3 to set up the climactic fifth set.
The ninth-seeded Croat raised up in the final set to break Tsonga at love for a 3-2 lead and served out the set, cashing in his fifth match point.
He won 6-4 6-4 3-6 6-7(3) 6-4 to advance to a semi-final against either top seed Novak Djokovic of Serbia or Spaniard Feliciano Lopez.
"It was a big mental fight, especially after losing that fourth set," Cilic, who served 29 aces, told the Arthur Ashe Stadium crowd in an on-court interview after the four-hour long quarter-final.
"Physically, it was very demanding."
(Editing by Andrew Both)
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