Bulgarian Dimitrov, seeded six, won the opening five games but made life difficult for himself before winning 6-3 3-6 6-3.
Medvedev, ranked 60, was a break up in the decider but Dimitrov hit back to win with something to spare.
He will face Spain’s 35-year-old Feliciano Lopez, who beat seventh seed Tomas Berdych 7-6(5) 6-7(1) 7-5, in Saturday’s semi-finals.