Austrian second seed Thiem, who ousted home hope Denis Shapovalov on Wednesday, faced two break points in the final game against the 2014 US Open champion before securing the triumph on a third match point.
Medvedev conquered his first match against Dominic Thiem to narrow down the head-to-head record against the Austrian 2-1.
Medvedev plans to play fewer ATP 250s in the next weeks: "At this moment I think my last 250 tournaments - where was it?"
"I lost things on the road so I just tried to add other things", Nadal added.
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With the win, Nadal made the Rogers Cup semis in Montreal for the first time since 2013 when he beat Canada's Milos Raonic in the final.
"I beat two incredible players", Thiem, who earlier beat Canadian young gun Denis Shapovalov, was quoted as saying by ATP World Tour.
The two-time Roland Garros finalist, who had never won a match in Canada over five years of trying before this week, said he did not have a chance as a hectic travel schedule caught up with him in the 56-minute rout.
Medvedev's best result at the Grand Slam tournaments is the fourth round of the 2019 Australian Open, he has won four ATP titles. "I was not on my 100 percent", he said.
Khachanov had little more problem with German Zverev, rattling through their quarter-final 6-3 6-3 in 74 minutes.
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Enfin, le Portugais a endossé ses responsabilités: "Le responsable des défaites, c'est moi, toujours. On l'accepte et nous devons continuer de travailler pour prendre confiance.
The loss ended a six-match winning streak for Thiem that went back to his run of form in winning the Generali Open in Kitzbühel last weekend.
Medvedev's semifinal opponent will be countryman Karen Khachanov. "There was more tension because you play at home".
"It's never easy to play a good friend, especially from the same country", Khachanov said.
Khachanov, ranked No. 8, blasted a ball out of the venue during Thursday's win over Auger-Aliassime in frustration at crowd noise during play, leading to loud boos. "We know each other pretty well".
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