The world No.1 gave Berankis no avenues into the match, winning 48 per cent (27-of-56) of his total return points, while also converting 88 per cent (21-of-24) of his own successful first serves into points.
Medvedev will meet Serbia's Miomir Kecmanovic in the semi-final after he pulled off an impressive 6-2 6-4 victory against American Brandon Nakashima.
Kecmanovic's ability to serve his way out of trouble was the deciding factor, saving four of five break-point opportunities, while Nakashima could only save six out of 10.
Canada's Felix Auger-Aliassime booked his spot in the second semi-final after defeating America's Steve Johnson 6-4 7-6 (7-3).
The 22-year-old winner, who now sits ninth in the world rankings, served up 17 aces in the contest, while Johnson, who is usually known for his serve, could only muster four.
Cameron Norrie was a straights-sets winner over Radu Albot, although he needed a second-set tie-break to get the job done in a 6-3 7-6 (7-4) triumph.
Meanwhile, Norrie's British compatriot Daniel Evans won a war of attrition against American Taylor Fritz at the Washington Open, pulling ahead 3-6 7-6 (8-6) 4-1 before the heavy favourite retired because of the oppressive conditions.
With rising temperatures, combined with spots of rain, the humidity threatened to claim a number of scalps as multiple players in Friday's (AEST) action took extended medical timeouts, before a serious downpour halted the rest of play.
The only other matches to reach their conclusion were Yoshihito Nishioka's 7-6 (7-2) 7-6 (7-1) win against Karen Khachanov, and a 6-3 6-7 (3-7) 6-4 triumph for Sweden's Mikael Ymer against Finland's Emil Ruusuvuori.
When play resumes, Australia's Nick Kyrgios will look to finish off Reilly Opelka after reaching the rain with a 7-6 (7-1) 2-1 lead, and Grigor Dimitrov won his first set 6-4 against Sebastian Korda.
Botic van de Zandschulp was the beneficiary of the break against Frances Tiafoe, winning the first set 6-4 before slowing down in a big way to drop the second 6-2, but he will get a chance to recharge his batteries before the decider.