Finland\'s Jere Lehtinen, right, attempts to score against Russia\'s goaltender Alexander Eremenko during the Ice Hockey World Championship semifinal match in Moscow, Saturday. (AP Photo/Sergey Ponomarev)
Finland will face Canada or Sweden on Sunday in its attempt for its second title and first since 1995. Russia, which has not won the title since 1993, will play for bronze.
Since the Soviet Union triumphed at the 1986 championships in Moscow, no host nation has ever lifted the trophy.
At 5:40 into overtime, Koivu skated into the slot to chase after a long pass from Tuomo Ruutu at the boards, pried Russian goaltender Alexander Eremenko's stick off the ice with his own and sliced the puck into the net.
Evgeni Malkin - favourite for the NHL's Calder Trophy as rookie of the year - delighted the home crowd on the power play 8:36 into the opening period, hitting a shot from the face-off circle that went over goalie Kari Lehtonen's stick.
The Finns equalized less than four minutes later, despite being short-handed, when Jukka Hentunen broke away, escaping Sergie Gonchar to beat Vasili Koshechkin with a backhanded flick.
Neither team was able to find the back of the net in the second and third periods.
Finland has been tested physically the last couple of days. Its hard-fought 5-4 quarter-final victory over the United States on Thursday went to a penalty shootout.
While the Finns had lost Petteri Nummelin, injured during the shootout with the United States, Russia lost tournament scoring leader Alexei Morozov to a knee injury.
It marked Russia's first ever defeat in five world championships in Moscow - 1957, 1973, 1979, 1986, 2007.
Russia had entered the game as favourite, undefeated and having posted a 5-4 win over the Finns in earlier round play.
The hosts had been looking to erase their humiliating performance from the 2000 worlds at St. Petersburg, where their star-studded roster crashed out in the second round, failing to get past even its weakest rivals.