American-born Turkish star Shane Larkin scored 26 points to go along with nine assists in the win at Moscow's Megasport Arena.
French guard Rodrigue Beaubois had 22 points and Serbian teammate Vasilije Micic added 16.
Anadolu Efes were leading 84-82 near the end of the game but CSKA Moscow's Italian guard Daniel Hackett was fouled when he hit a three-pointer.
Hackett scored from the free throw line to make it 86-84 for the home team.
But Micic scored on a driving layup to tie it at 86 with less than 1 second left in the fourth quarter.
Micic's solo effort took the game to overtime.
The extra period was also tight.
Beaubois hit a three-pointer to give Anadolu Efes a 97-94 lead with 49 seconds to go.
Then Krunoslav Simon scored on a free throw to increase the lead to 98-94.
Lithuanian forward-guard Marius Grigonis scored on a three-pointer to give hope to CSKA Moscow with 30 ticks left.
Guard Alexey Shved was free in the corner but missed on a three-point attempt.
The Russian team hurried to foul Adrien Moerman to stop the clock. The referee decided it was an unsportsmanlike foul. Moerman missed both free throws but Anadolu Efes retained possession as Beaubois was fouled.
Beaubois went to the free throw line to score on the front end of his first free throw to make it 99-97.
The French guard missed his second to start the clock with just 0.7 seconds remaining.
But time ran out when Hackett took a defensive rebound, giving the Turkish club the narrow win.
Hackett was the game's top scorer with 31 points. Grigonis had up 20 for the Russian powerhouse. CSKA Moscow forward Will Clyburn added 19.
Eighth-place Anadolu Efes had their 11th win. They previously lost 11 games.
CSKA Moscow are in the fifth spot in the EuroLeague standings. They have a 13-9 record.
Real Madrid are 18-3 to lead the league.
Next up for Anadolu Efes will be France's LDLC ASVEL Villeurbanne at Istanbul's Sinan Erdem Sports Hall on Friday.
