Anadolu's first victory in the second group stage of the competition completed a memorable round for Turkish teams as Fenerbahce also beat Panathinaikos 77-72 and Galatasaray defeated Bayern Munich 72-66.
Milan looked set for a valuable road win after forward Keith Langford buried the first of two free throws to secure a 60-58 lead with three seconds left.
But the American missed the second free throw and Croatian forward Planinic had the passionate home fans in raptures as he hurled the ball in from long range.
"It is not strange for me to make this kind of shot because I have done it before," Planinic told the Euroleague's wesbite. "My only thought after collecting the rebound was to shoot as soon as I could and I just turned around and threw the ball.
"I hope this shot changes things for me and for Efes," said Planinic whose patchy performances this season have gone hand in hand in hand with his team's inconsistent form.
It represented a lifeline for Efes in their bid to reach the knockout stages after losing their opening three games in the Top 16 that features two pools of eight teams.
The top four from each section advance to the best-of-five quarter-final playoff series and the winners qualify for the May 16-18 Final Four in Milan.
Dark horses Fenerbahce grabbed their first Top 16 win of the season after Lithuanian forward Linas Kleiza nailed a pair of long-range shots to settle a rollercoaster game against Panathinaikos.
It also took the pressure off Serb coach Zeljko Obradovic in an emotional clash with his former club.
Obradovic, who steered Panathinaikos to five Euroleague titles and 11 Greek championships in 13 years with the team, underlined how difficult it was for him to face some of the players who helped fill his trophy cabinet.
"It was very tough for me to play against my former team and very strange to see (playmaker Dimitris) Diamantidis and (forward Antonis) Fotsis on the other side," said the coach.
"They are in my heart but I am at Fenerbahce right now and I will keep fighting until the end for my team."
Holders Olympiacos Piraeus, who won all 10 games in the preliminary group stage, have a 2-2 record in the Top 16 after Matthew Lojeski helped them to an 89-70 victory at Spaniards Laboral Kutxa Vitoria with a game-high 17 points and six rebounds.
CSKA Moscow ended Real Madrid's 31-game unbeaten run in all competitions by beating them 85-71, Barcelona edged out Unicaja Malaga 83-79 in an all-Spanish clash and Lokomotiv Kuban Krasnodar ground out an 80-75 home win over Zalgiris Kaunas.