Medvedev came through three tough matches in the group stages but found progress a lot more smooth against debutant Casper Ruud in the semi-finals.
The young Norwegian battled to victory against Andrey Rublev on Friday to book his spot in the last four but second seed Medvedev was far too solid, the Russian winning 6-4 6-2.
Medvedev will take on either Novak Djokovic or Alexander Zverev in the final on Sunday.
Salisbury and American Rajeev Ram have had another brilliant season, capped by their second grand slam title at the US Open, and they came from a set down to claim a 4-6 7-6 (3) 10-4 victory over top seeds Nikola Mektic and Mate Pavic in Turin
Having let a 7-1 lead slip in the deciding tie-break at the same stage last year against Jurgen Melzer and Edouard Roger-Vasselin, Salisbury and Ram ensured they did not make the same mistake this time.
Salisbury is looking to follow in the footsteps of 2016 singles champion Andy Murray who is the only previous British winner of the event.
Salisbury was unaware of his history-making moment, saying: “It obviously makes it extra special. Hopefully there will be a lot more.
“It’s a big achievement. Obviously really, really proud of it. But I don’t want to be just a Brit getting to the final, I want to be the one winning it. We’re enjoying the win right now, but we’re going to be back to business soon, really focusing on the match tomorrow and getting the title.”