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Medvedev overcame a shaky start and converted four break points while dropping serve just once to secure a spot in the last eight in Miami, where he has yet to drop a set.
Koepfer was the early aggressor, using a pair of holds at love to build a 4-3 lead in the first set, but Medvedev forced a tie-break in which he lost the opening four points before finding his way.
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With a frustrated Koepfer suddenly losing control of his forehand, Medvedev enjoyed much smoother passage through the second set where he broke twice to build a commanding 4-0 lead before closing out the match with a break to love.
"Sometimes it happens when you lose the first set the way he lost...it brings your (opponent's) energy down and I knew that I had to use it (to my advantage) in the beginning of the second set - that's where it was the most important," Medvedev said after collecting his 350th career win.
The Russian will face Chile's Nicolas Jarry, the 22nd seed, in the last eight after the latter defeated seventh seed Casper Ruud of Norway 7-6 6-3.
Australian Open champion Jannik Sinner, seeded No 2, improved to 19-1 in matches this year with a 6-4 6-3 victory over Christopher O'Connell to also reach the quarter-finals.
In the middle of the match, commentators said a spectator had fainted in the stiflingly hot temperatures and play was stopped. Sinner came to the sideline and passed up drinks and towels to the person.
In other early action, unseeded Czech Tomas Machac beat Italy's Matteo Arnaldi 6-3 6-3 to reach his first Masters 1000 quarter-final.
Top seed Carlos Alcaraz, who beat Medvedev in the Indian Wells final nine days ago, was due to meet Italian 23rd seed Lorenzo Musetti in the late game.
Azarenka books semi-final place after gruelling epic vs Putintseva
On the women's side, two-time Grand Slam champion Victoria Azarenka defeated Yulia Putintseva 7-6 1-6 6-3 to make the semi-finals.
It was the second time that Azarenka, 34, has made it to the semi-finals this season after going that far just once in 2023. She has won the Miami Open three times.
Playing on Stadium Court, the players also had to deal with a 53-minute delay due to a power outage, before playing out an epic first set which lasted 90 minutes.
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