In a fiercely-contested match, Cilic's powerful groundstrokes gave him a 5-3 lead in the first set but Mannarino fought back to force a tiebreak, which the Frenchman led 4-2 before losing five of the next six points.
Mannarino took advantage of Cilic's weaker backhand to take a 5-3 lead and although he lost his next service game, the 29-year-old broke the Croatian's serve to win the second set.
Mannarino won seven straight games to beat world number five Cilic and secure his first win over a player ranked in the world's top five.
Goffin and Schwartzman both saved break points as the first set went to a tiebreak and with the score tied at 4-4, the eighth-seeded Schwartzman made three straight errors to gift his opponent the set.
Belgian Goffin served for the match in the 10th game of the second set but a forehand error enabled Schwartzman to break back and force another tiebreak.
Goffin sealed victory on his second match point to extend his winning streak to eight matches after his Shenzhen Open triumph last week.
He moved up to 10th in the Race to London rankings and is still in contention for a place in the eight-man season-ending tournament.
(Reporting by Aditi Prakash in Bengaluru, editing by Ed Osmond)
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