Top seed Djokovic, playing the week before Wimbledon for the first time since 2010, brushed aside Medvedev 6-4, 6-4 and will face second seed Gael Monfils in Saturday's final, after the Frenchman edged out fellow countryman Richard Gasquet 6-2, 6-7 (7/9), 7-6 (7/4).
"I am enjoying undoubtedly this week and the fact that I'm in the finals obviously makes it even better," Djokovic said. "I'm really glad that I'll be able to fight for the trophy tomorrow.
"I haven't had too many opportunities in the last 10 to 12 months. I haven't had too many finals, so it's actually a good feeling.
"It comes in the right moment, because I need to build a level of confidence; the matches won obviously help that.
"Coming into Wimbledon, I'm hoping that I can continue in this path of raising the performance and the quality of tennis."
But British hopes were extinguished as a back injury forced Konta to pull out of her clash against Karolina Pliskova, with Heather Watson then falling in three sets, losing 6-2, 3-6, 7-5 to former world number one Caroline Wozniacki.
Seventh-ranked Konta suffered a nasty fall in Thursday's quarter-final win over world number one Angelique Kerber and was subsequently diagnosed with a thoracic spine injury.
Djokovic's match on Friday was delayed by rain, but the Serb gained control when he broke his fellow wild card to take the opening set, with an early break in the second steering him through to just a third final of the year.
The world number four is seeded second at Wimbledon on the strength of three titles at the All England Club but is looking for his first trophy since winning at Doha in January.
As for Konta, the 26-year-old is Britain's best hope of ending a long wait for a women's Grand Slam champion that dates back to Virginia Wade winning the 1977 Wimbledon singles title.
Konta, who won the prestigious Miami tournament earlier this season, had originally been passed fit to play on Friday but decided to pull out to avoid risking further damage.
"We made the decision based on the fact I'm still quite sore through my thoracic spine. Next week is Wimbledon but I make decisions for my health," said Konta.
"I didn't sleep too well but I heard that's normal. It just didn't feel quite right. The most important thing is I rest well for the next 24 hours and then we'll see.
"Right now my priority is to look after myself, quite honestly.
"I'm looking to just make sure that I'm in a good place physically and health-wise to be able to compete next week, but more importantly just to be in a good place health-wise."
Konta will hope the pain diminishes over the weekend as she is due to play Taiwan's Hsieh Su-wei, her conqueror at Roland Garros last month, in the first round at Wimbledon on Monday.
Watson has dropped to 126th in the rankings but put up a stiff challenge against 2009 Eastbourne champion Wozniacki.
However, the Briton was broken as she sought to force a final set tie-break, sending Wozniacki through to a second final on the English south coast.