Barca suffered only the second defeat of the season in Glasgow, the other a 2-1 loss to arch-rivals Real Madrid in the second leg of the Spanish Super Cup in August.
However, they remain top of the group on nine points, two ahead of Celtic in second. Benfica have four points and Spartak Moscow are bottom on three after the Portuguese side won 2-0 at home to the Russians on Wednesday.
"We are still in first place and we are relaxed," Barca's World Player of the Year Lionel Messi, who netted in added time and earlier struck the frame of the goal, said.
"I think we did everything well, we got into goal-scoring positions but the ball just didn't go in," added the Argentine, who became a father for the first time when his son Thiago was born on Friday.
"They were comfortable closing ranks in defence, waiting for a chance to counterattack, a corner or a freekick and that's how they managed to score the two goals."
Barca coach Tito Vilanova and his players were generally gracious in defeat and there was a sense they had even enjoyed the electric atmosphere at Celtic Park despite the defeat.
"It's a marvel, an example," Xavi said.
Celtic's victory ended Barca's run of 17 away games in the Champions League group stage without losing.
They last fell at Chelsea, 1-0 in October 2006, and had since won 11 and drawn six.
They have another chance to seal their passage to the last 16 away to Spartak on November 20 before they host Benfica on December 5.