World No.1 Sinner is just one victory away from becoming only the second man after Novak Djokovic to win all nine Masters 1000 titles, the Serb having won each tournament at least twice.
Sinner led 4-2 in the third set over the Russian when the match was suspended late on Friday.
But the Italian needed just 15 minutes on Saturday to wrap it up, sealing victory on his third match point after Medvedev fended off two on his serve.
"It was a very different challenge and a tough challenge," Sinner said.
"Usually, during the night, I don't struggle to sleep but this time it was not easy.
"You are in the third set, nearly done, but you still have to show up again and you never know what is happening. It is like the start of the match as there are nerves again.
"I am very happy with how I handled this situation and that I am back in the final."
Sinner is attempting to become the first Italian man to raise the Rome trophy since Adriano Panatta 50 years ago.
Panatta himself will present the trophy on Sunday, with Italian President Sergio Mattarella also slated to attend the men's final.
But though he's been dominant, Sinner had appeared fatigued on Friday as Medvedev stepped up his game and started running him around the court with drop shots and groundstrokes to the corners.
After several points Sinner bent over in apparent exhaustion and leaned on his racquet for support. He also had his right thigh treated by a trainer midway through the second set.
Sinner came back from a 3-0 deficit only to be broken again in the 12th game and concede the set to Medvedev, but a bad bounce helped the Russian to get broken early in the third as the Italian took control for good.
Ruud knows he faces the tallest of orders. "Jannik is chasing history," the Norwegian said.
"Last year, he really routed me here on the same court, so of course I'm looking for revenge. But at the same time I realise that he's an incredible player and a unique talent."
Sinner, who lost last year's final to Carlos Alcaraz, hasn't lost any match since February 19 in the Qatar Open quarter-finals. He's won 28-straight matches and a record five successive Masters titles.
