Khachanov will face Jakub Mensik in the final after the Czech native beat Frenchman Gael Monfils 6-4, 1-6, 6-3.
Khachanov won the first four points of his first-set tiebreaker but let Popyrin right back in it, leading to a 6-5 deficit. The Russian saved the first of three set points he faced in the tiebreaker, but he did not finish the job until his fifth set point at 13-12.
Despite Popyrin having a 35-23 advantage in winners, Khachanov had just four unforced errors to Popyrin's 14 and won 55.6 percent of the total points (79 of 142).
Mensik, who upset top-seeded Andrey Rublev of Russia in the previous round, saved nine of 12 break points against Monfils. Mensik is ranked No. 116 in the world.
Mifel Tennis Open
Eighth-seeded Jordan Thompson of Australia is one win away from his first career ATP Tour title after upsetting top-seeded Alexander Zverev of Germany 7-5, 4-6, 7-6 (2) in the semi-finals at Los Cabos, Mexico.
Thompson has been to two previous tournament finals, both at Hertogenbosch, Netherlands, in 2019 and 2023. He needed seven match points to get past Zverev, five on the German's serve at 5-6 in the third set and the other two in the decisive tiebreaker.
Thompson's title-match opponent will be fourth-seeded Casper Ruud of Norway, who ousted second-seeded Stefanos Tsitsipas of Greece 6-4, 7-6 (4) in the first semi-final. Ruud is chasing his 11th championship on tour, most recently having prevailed at Estoril, Portugal, last year.
Rio Open
Two Argentine players beat Brazilian opponents and a third Argentine player joined them in earning quarter-final wins in Rio de Janeiro.
Fifth-seeded Sebastian Baez got past Thiago Monteiro 6-4, 1-6, 6-2 and qualifier Mariano Navone got past Joao Fonseca in the two Argentina-Brazil matchups. Fonseca, 17, had never won a match on the ATP Tour before winning twice in Rio this week.
Baez's semi-final opponent will be countryman Francisco Cerundolo, the fourth seed, who rallied past Serbia's Dusan Lajovic 3-6, 6-4, 6-4. Navone's semi-final foe will be second-seeded Cameron Norrie of Great Britain, who moved on with a 6-1, 3-6, 6-2 victory over Brazil's Thiago Seyboth Wild.
