Russia's Medvedev quelled a late resistance from Kovalik for a 6-4 6-1 win that came in one hour and 28 minutes.
Nevertheless, it turned out to be superb tournament for Kovalik, who came through qualifying and scalped the biggest player of the event, world number six Marin Cilic.
The Russian was playing much better than his first round effort. He not only served big as he fired eight aces and had sting in his shots, he smartly played drop shots to catch Kovalik by surprise on several occasions.
Both the players were unhappy with many calls during the match but Medvedev chose not to make an issue out of it.
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"It's part of the game. The conditions were same for both of us," he said adding that the Russian players needed more support from the national Federation.
He next faces Israel's Dudi Sela, who battled past Spanish third seed Albert Ramos-Vinolas 7-5 6-4.
Also making it to the last-four was French fifth seed Benoit Paire, who had an easy outing there, beating Aljaz Bedene 6-3 6-0 as the British struggled with a back problem.
Kovalik finally got on board in the fifth game but not before saving a breakpoint in that game.
Struggling to find a way out, Kovalik several times looked at his coach Boris Borgula sitting in the stands, which apparently made him look more helpless against a relentless Medvedev.
Medvedev felt the heat for the first time in the match when the Slovak started to return better and found a forehand winner at 40-40 to earn a breakpoint but the Russian managed to save with his big serve.
The Slovakian started to play much better in the second set and was rewarded for perseverance. He returned well and had more power behind his shots.
Medvedev made some unusual unforced errors and a third consecutive backhand error meant that Kovalik got first break of the set in the fifth game to lead 3-2.
Kovalik's bad temper got to him as he allowed the Russian to get the break back when he double faulted at deuce and lost a long rally by netting a backhand. Kovalik threw his racquet on to the advertising boards out of frustration.
Meanwhile, second seed and world number 14 Roberto Bautista Agut came from behind to edge past seventh seed Russian Mikhail Youzhny 2-6 6-4 6-4 in the quarterfinals.
Agut broke the Russian in the ninth game of the deciding third set on the third breakpoint. Youzhny had two chances to break Agut in the 10th game to make it 5-5 but squandered both breakpoints.
The Spaniard next faces fifth seed Benoit Paire from France.