Kilmarnock manager Allan Johnston and midfielder Jamie Hamill accepted referee Willie Collum had no option but to send off skipper Manuel Pascali for a professional foul at Celtic Park.
The Italian defender, 33, was shown a straight red for bringing down Celtic striker Stefan Scepovic after 34 minutes when the match was goalless.
"It looked as if he switched off," said Johnston after the 2-0 league defeat.
"Manuel is devastated. He feels he has let the players down."
John Guidetti scored from the resultant free-kick and Scepovic added a second after the interval, the win lifting Celtic above Aberdeen and Kilmarnock into fourth spot in the Scottish Premiership.
However, the Ayrshire side are only three points off Hamilton and Inverness at the top.
Kilmarnock had played with confidence prior to the red card, with their midfielders keen to get on the ball to make passes.
"I thought in the first 35 minutes we looked dangerous and created a few chances but the sending off completely changed the game," said Johnston.
"I haven't seen the incident again but I don't think we can complain about it. It was the big decision in the game.
|Kilmarnock midfielder Jamie Hamill|
|"It's one of those ones - sometimes you have to make a professional foul."|
"Manuel is gutted it happened in such a big game. He has been brilliant for us since the start of the season.
"It was not the way we came to play. You could see that in the opening 35 minutes we tried to pass the ball whereas in the second half it was two banks of four and we tried to hit them on the counter-attack.
"We made it hard for them to break us down. It was always going to be difficult but I think the players were great. It's not easy to be down to 10 men for so long at Parkhead."
Hamill echoed the views of his manager, suggesting the referee had "got it right".
"It's one of those ones - sometimes you have to make a professional foul," he said.
"They would probably have gone 1-0 up if he [Scepovic] had gone through to finish.
"It was a difficult one to come back from, especially when we conceded the goal straight from the free-kick that Manny gave away."