Rublev said his sixth career meeting with Medvedev reminded him of his encounter with the world number five at the 2020 US Open quarter-finals, where he lost in straight sets, and he was determined to overcome his poor start.
But having shaken off the disappointment of squandering several chances to take the opening set, Rublev hit back with 25 winners in the next two and held his nerve in the deciding tie-break to seal the victory on his fifth match point.
"I was thinking about the US Open, because when we played in the quarter-finals in 2020, I think I was 5-1 in a tie-break, also something like 6-2, but I was leading something similar in a tie-break," Rublev said.
"But when I lost that set, I couldn't play anymore, and I was thinking it can't happen (again) I have to change something. I have to keep playing, because if I win one set, even if I lose in three sets, it's still good, because here you have a group.
"So you have to fight for every point, every game, and I was able to win. The final tie-break, the rallies that we had there were crazy. The last rally, we had I don't know, 30 shots."
Rublev said he was cramping towards the end of the contest before winning with a 37-shot rally on match-point, after which he dropped to the floor in delight.
"I was thinking 'one more, one more. You have to keep playing'," Rublev, who also beat Medvedev in the Cincinnati semi-finals last year, said.
"For sure, he felt the same, 'so just keep playing if you have the chance, go for it'. In the end I was able to win."
Stefanos Tsitsipas (24) and Novak Djokovic (35) face off in the other Red Group match later on Monday.
