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Medvedev had never beaten Bautista Agut prior to Wednesday's showdown – the world number two upstaged in the 2017 Maharashtra Open final and last year's Western & Southern Open quarter-final.
The Russian star was once again unable to solve the Bautista Agut riddle, upstaged 6-4 6-2 at the ATP 1000 tournament midweek.
Bautista Agut – the seventh seed and oldest player remaining in the draw – broke for a 5-4 lead in the opening set and never looked back against the 2021 Australian Open runner-up.
The only player in the last eight to have reached the quarter-final stage previously, Bautista Agut continued where he left off in the second, making Medvedev uncomfortable as he booked a spot in the semi-finals.
Spanish veteran Bautista Agut will face Jannik Sinner in the semi-finals of the Miami Open.
Italian sensation Sinner proved too good for Alexander Bublik 7-6 (7-5) 6-4 in the day's early match.
Sinner, 19, became just the eighth teenager to reach the semi-finals in the tournament's history – five of those men have gone on to top the ATP rankings, Andre Agassi (1990), Lleyton Hewitt (2000), Rafael Nadal (2005), Novak Djokovic (2007) and Andy Murray (2007).
Sinner drew high praise from a stunned Bublik post-match in Miami.
"He's not [human]. That's a fact," said Bublik. "I asked him if he's a human or not, because for me, it's very surprising that the guy at his age has this mental toughness that many, many other players don't have.
"I called him a robot a couple of times during the match, but I do it in a very sincere way because he's a really, really a great player."