In the semifinals, the 33-year-old Spanish star will face No. 10 Roberto Bautista Agut of Spain or No. 16 Gael Monfils of France.
The women's Rogers Cup bracket is being played in Toronto. The eighth-seeded Russian, who improved his season-long record to 30-15, has been on a roll this summer, first breaking into the ATP Top 10 on July 15.
Second seed Dominic Thiem and third-seeded Alexander Zverev suffered surprise defeats in Friday's ATP Montreal Masters quarterfinals, leaving top seed Rafael Nadal a relatively clear field in defending his title.
The pair gave fans an incredible rally in the fifth game that included a between-the-legs shot from Nadal before Fognini deadened the ball at the net to win the point. However, the Austrian couldn't break him in the next game as Medvedev held his serve to win the set 6-1.
Nadal broke serve to start the deciding set and never looked back as he took a measure of revenge against Fognini, who defeated Nadal in the final of the Monte Carlo Masters in April.
"Everybody knows he's one of the big talents on tour", said Nadal, who lost to Fognini in the third round of the 2015 U.S. Open.
Nadal initially had a more tricky time in his quarter-final against Fognini, struggling with his serving accuracy in a first set where the Italian broke him twice. He also won the event here in 2005, and in 2008 in Toronto.
The 23-year old Russian prevailed 6:3, 6:1 in 56 minutes.
"I beat two fantastic players", Thiem, who earlier beat Canadian young gun Denis Shapovalov, was quoted as saying by ATP World Tour. I heard a lot about it that there were good food and stuff but never saw it. "It saved me a lot of energy".
The victor past year in Toronto for his fourth title in the event, Nadal won his 380th ATP Masters 1000 match, a day after breaking a tie with Roger Federer for the tour record.
"The battery was empty", said the 25-year-old. "I was not on my 100 per cent", he said.
The sixth-seeded Khachanov, who downed hometown favourite Felix Auger-Aliassime on Thursday to advance, also made quick work of Zverev and should be equally well-rested Saturday. Maybe against other players it would have been a closer match.
Zverev, the third seed and No. 7 player in the world, slammed his racket in frustration in the first set, which elicited a negative reaction from the crowd.
Looking to watch (and bet) on the Dominic Thiem v Daniil Medvedev live streaming online? The crowd politely applauded his early breakthrough Friday, but support grew louder after Zverev's outburst.
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