"No Davis Cup match is easy and you don't take anything for granted, rankings get thrown out the window," said Australia captain Lleyton Hewitt. "We knew they were going to be tough opponents. I'm really proud of the boys."
Popyrin edged Virtanen in the first set, saving a set point when trailing 6-5 and forcing a tiebreaker that he clinched when his opponent slapped a forehand long.
Virtanen was Finland's hero in its upset of defending champion Canada on Tuesday, winning both his singles and doubles matches. But his serve let him down against Popyrin in the second set when his fifth double-fault cost him a break.
"(That was) probably the biggest win of my career so far," Popyrin said. "To win a match that means so much for us is an honor and something I will never forget."
De Minaur converted five of 18 break chances against Ruusuvuori. He ended the match by landing a backhand winner on the sideline.
Australia has the second-most Davis Cup titles, with 28 to 32 for the United States, but its last success came in 2003.
Before beating Canada to reach the semifinals for the first time, the Finns had overcome the United States and Croatia at the group stage.