After Roger Federer suffered a shock defeat on Wednesday, Zverev, who reached the semi-finals of the French Open last week, also exited in the second round when he lost 7-6 (7/4), 3-6, 6-3 to France's Ugo Humbert.
Italian Open, Grigor Dimitrov, Alejandro Davidovich Fokina, Roberto Bautista Agut, Tommy Paul, Nadia Podoroska, Serena Williams, Laura Siegemund, Madison Keys, Sloane Stephens
Sinner, who reached last year's French Open quarter-finals, defeated Bautista 5-7, 6-4, 6-4 to become only the fourth teen to reach the men's final in the 36-year history of the ATP Miami Masters.
Barty, who is assured of retaining her No 1 ranking next week, will play for the title Saturday against the winner of the second semi-final between No 8 Bianca Andreescu and No 23 Maria Sakkari.
Seventh-seeded Spaniard Roberto Bautista Agut beat Medvedev 6-4, 6-2, for his third win in as many meetings with the Russian.
Miami Open: Ashleigh Barty reaches semis after hard-fought win over Aryna Sabalenka; Daniil Medvedev into last eight
Meanwhile, Jannik Sinner downed Emil Ruusuvuori 6-3, 6-2 to advance to the last eight.
Medvedev failed to convert three match points in the second set after taking a 5-2 lead. An hour later the Russian was still playing, and on a humid afternoon with temperatures in the mid 80s, he paid a price.
Making his tournament debut this week, the 19-year-old Sinner fired 16 aces during the two-hour 24-minute battle to set up a last-eight encounter with Lorenzo Sonego or Aslan Karatsev.
Their clash in the Gulf desert nation was characterised by hot, dusty wind that slowed down their exchanges but world number 42 Basilashvili bided his time to snatch the first set in a tiebreak.
Bautista Agut eased past Russia's Andrey Rublev, ranked eighth in the world and the defending champion, with a dominating performance, 6-3, 6-3
Looking ahead, Federer also said that he had not taken decisions on tournaments before Wimbledon or the Tokyo Olympics but hinted that he would likely be at Roland Garros.
World number three Medvedev was the top seed for this week's tournament following his run to the Australian Open final but was dumped out 7-6 (7/4), 6-4 in just over 90 minutes by Serbian Lajovic, currently ranked 27th.
Open Sud de France: Belgium's David Goffin wins first title in over three years after beating Roberto Bautista Agut
Goffin, the 30-year-old world number 15, saw off top seed and 13th-ranked Bautista Agut 5-7, 6-4, 6-2.
Humbert trailed 2-5 in the second set and faced the match points at 3-5, then turned things around to clinch a 6-7 (4), 7-6 (5), 7-6 (5) win after nearly three hours.
The 20-time Grand Slam champion was due to take the court against Australia's top-ranked player Alex De Minaur, but opted to rest instead as he prepares for the Australian Open next week.
Alexander Zverev beats Alejandro Davidovich Fokina to set up Cologne Open final against Felix Auger-Aliassime
Felix Auger-Aliassime defeated second-seed Roberto Bautista Agut 6-3, 1-6, 6-3 in the semi-final to reach the summit clash.
Alexander Zverev beats Fernando Verdasco to reach quarters in Cologne; Daniil Medvedev loses in St Petersburg
Alexander Zverev was playing indoors for the first time this year after a mixed record on indoor hard courts in 2019.
Dominic Thiem, the losing finalist in the Australian Open this year, battled to a 6-2, 6-2, 3-6, 6-3 victory in the late match at an empty Arthur Ashe stadium.
US Open 2020: Matteo Berretini's loyal supporter makes presence felt, Vasek Pospisil surprises himself
No fans are allowed inside the gates at the US Open, but Matteo Berrettini's most loyal supporter found a way to offer encouragement by yelling from outside a fence near Court 17.
US Open 2020: Serena Williams beats Sloane Stephens to reach last-16; Kristina Mladenovic-Timea Babos withdrawn after quarantine notice
Unranked Bulgarian Tsvetana Pironkova caused her second upset of the week at US Open, knocking out 18th seed Donna Vekic in straight sets.