Last year's runners-up Maccabi Tel Aviv took a 2-1 lead against holders Panathinaikos Athens after grinding out a 65-62 home win over the Greek side in front of a fervent crowd in the Nokia Arena.
Barcelona, the 2010 winners, had won the opening two games in their Palau Blaugrana last week. They trailed throughout the first half in Kazan on Tuesday before playmaker Juan Carlos Navarro inspired them to an impressive comeback.
The all-time leading scorer in Europe's premier club competition buried four three-pointers to top the charts with 21 points, finishing as Barcelona's only player in double scoring digits.
"We expected a difficult road game against Unics since we played two tough matches in Barcelona and we won after playing much better in the second half," Navarro told the competition's official website (www.euroleague.com).
"Unics have had a fantastic first season in the Euroleague, they reached the top eight and played well all season long," added Barcelona coach Xavier Pascual.
Maccabi, beaten by six-time Euroleague winners Panathinaikos in last year's final, seemed to be cruising after carving out a 41-28 lead at halftime.
Long-range efforts by Panathinaikos forward Stratos Perperoglu forced a dramatic finale and his team mate Nick Calathes missed a chance to send the game into overtime when his three-point attempt on the buzzer came off the rim.
Playmaker Yogev Ohayon led the winners with 12 points, one more than shooting guard Devin Smith who also had nine rebounds.
CSKA Moscow, who are 2-0 up against Euroleague debutants Bilbao, can seal their own Final Four berth when they visit the Spanish side on Wednesday while Italians Montepaschi Siena head to Olympiakos Piraeus tied 1-1 with the 1997 winners.