Mulgrew claimed the referee had been intimidated into sending Brown off by the Barcelona players.
The referee had first taken the yellow card out of his pocket in response to the initial foul on Neymar, but appeared to have a clear view of the kick that followed and changed his mind amid protests from the visitors.
"It's hard to go down to 10 men so early in the second half," Mulgrew said. "It was always going to be an uphill battle. I thought we defended really well and had a few chances. If we'd have taken them it would have been different."
The Scotland striker exonerated his captain, claiming Brown had not intended to connect with Neymar's arm while the Barcelona forward was down.
"I think he's just taken a little step, and (the referee) went with the player's reaction. It was a bit harsh. I don't think he was convinced it was a sending off.
"He has got to be sure. Big decisions like that change the match and it killed us. Eleven men against Barcelona is hard enough so it definitely changed the match."
The result leaves Celtic bottom of the group, but Mulgrew insisted the fight to reach the next round would go on.
"We've got to keep believing and putting in performances," he said. "If we can take our chances we can definitely qualify."
Celtic manager Lennon was also upset by the red card, insisting his side had been better up to that point.
"I'm not convinced," he said ."Any contact that was made was minimal. It's in front of the referee but there's a lot else going on out there. It changed the game."
However, Lennon had plenty of positives to reflect on.
He added: "I can't speak highly enough of the players. We've had two chances right before they score. There's a great save from Valdes and then Charlie should score if I'm honest.
"I don't remember (goalkeeper Fraser Forster) having a lot to do until the game got stretched at the end.
"We were going for the equaliser late on but until 75 minutes or so he didn't have a lot to do."