The Black Eagles took the lead on the half-hour mark through Ernest Muci, but their opponents fought back to secure a semi-final spot.
Despite the defeat, the former Manchester United manager has refused to blame his team for the quarter-final exit.
Solskjaer told Besiktas website: “We played well for the first 35 minutes. We were even ahead of our opponent in terms of shots.
“We controlled the game and had some chances. The goal that came after the red card changed everything. Then, an unlucky hand movement led to a penalty.
“Sometimes in football, these small differences and details determine things. These small differences and details also worked against us.
“However, I cannot blame my players. They all fought. They tried to fight until the last moment. We had opportunities even when it was 3-1.
“So, I cannot blame my players for not making an effort or not giving their 100 percent. They did their best.”