Finland trailed in both regulation and first overtime but tied it late both times to remain in the game.
Elsewhere, Ricky Rubio and Marc Gasol scored 15 points apiece to lead defending champion Spain to an overwhelming 89-53 victory over winless Poland in Celje.
Five Spanish players scored in double figures to secure their team's third victory in four games. Spain led 49-13 at halftime and led by as many as 48 points before slowing down in the second half.
Bosnia-Herzegovina beat Macedonia 62-54 and Britain eliminated Germany with an 81-73 victory.
Bosnia is now 2-2 while Macedonia fell to 1-3 as the semifinalist from two years ago is on the verge of being eliminated. Top three teams from each group advance.
In Ljubljana, Devon van Oostrum's 3-pointer with 50 seconds left gave Britain a seven-point lead and secured the victory.
Britain, which is still not certain to advance, got 23 points from Andrew Lawrence and 21 points and 11 rebounds from Myles Hesson and is 2-2. Tibor Pleiss had 20 points and 14 rebounds for Germany, which fell to 1-3.
In Koper, Hanno Mottola sank a 3-pointer for Finland with three seconds remaining in regulation to force overtime. Mottola put in a tip-in and Gerald Lee's lay-up again tied the game at the end of the first overtime.
Sergey Monya's 3-pointer made it 82-82, before Lee dunked and made one of two free throws for a three-point Finland lead. Monya then missed one free throw and Sasu Salin made on for Finland. Russia called a timeout with six seconds remaining but Monya came out to miss a 3-point attempt and Finland had the win.
The defeat means exit for Russia, the bronze medallist at the London Olympics last year and at the Europeans two years ago.
The Russians were without Andrei Kirilenko and several other stars.
Shawn Huff led Finland with 20 points and nine rebounds, while Salin and Tuukka Kotti had 12 points and 10 rebounds each. Finland also got 11 points and 11 assists from Petteri Koponen.
In Jesenice, Mirza Teletovic had 22 points and Zackary Wright pulled down 12 rebounds for Bosnia, which outscored Macedonia 12-2 in the last five minutes. BoMcCalebb led Macedonia with 19 points, but made some bad plays late in the game, while Pero Antic had 17 points.