The home favourite was leading 6-2 5-7 4-2 and once again looked untouchable on home clay, racing through the opening set in just over half an hour while losing only two points on serve.
But Medvedev raised his level in the second set to force a decider against Sinner, who appeared increasingly hampered physically as the Russian became the first player to take a set off the top seed in the tournament.
Sinner responded immediately in the third, securing an early break to ignite the Campo Centrale crowd and seize control of the match, despite taking a medical timeout after showing signs of cramping.
Play was then halted with rain pouring down in Rome and organisers later announced the match would resume on Saturday.
Sinner is chasing a sixth consecutive Masters 1000 title and the only one missing from his collection, adding to recent triumphs in Paris, Indian Wells, Miami, Monte Carlo and Madrid.
The winner will face Norway's Casper Ruud in the final after the Norwegian cruised to a dominant 6-1 6-1 victory over Luciano Darderi.
Ruud reached his first Rome final by taking advantage of a fatigued Darderi, who was coming off a marathon quarter-final victory in the early hours of Thursday.
The 23rd seed is seeking his second Masters 1000 crown after winning the Madrid Open last year.
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