With his side 2-0 up and in command of the match, McCulloch decided to contest a high ball against Hearts striker Osman Sow by lunging into his opponent, catching Sow in the jaw with an elbow.
Once play had stopped, referee Bobby Madden showed no hesitation in brandishing the red card.
In an interview with BT Sport a contrite McCulloch said: “I felt as though I was caught in no-man’s land; if he won the header, there would have been a runner in. There was no intent whatsoever, I can say that honestly, but obviously I’m disappointed.
“But the bigger picture is the boys got three points. It was a great day for the fans and the players and we’re keeping the momentum going and I think that’s all that matters.
“I’ve apologised to all the management, all my team-mates, I feel as if I’ve let myself down. But, as I’ve said, the big positive is it’s a great result for the fans, the boys dug in and showed great character and I’m delighted with that.”
Stuart McCall insisted he did not see the incident in question, although the Rangers manager did admit that McCulloch apologised to the team. McCall said he would deal with his skipper after the elation of the crucial 2-1 victory died down.
He said: “I missed it at the time because I was concentrating on [Genero] Zeefuik going through. I spoke to Bobby Madden coming off and he said that he’d caught him with his elbow. Lee came in afterwards and apologised to the team so it must have merited a red card.
“I’ve not had time to think about it. I’ll sit down and look at it myself and speak to him. At this moment in time I’m quite delighted with the level of performance of the players.” Hearts boss Robbie Neilson, meanwhile, revealed prior to kick-off that full-back Adam Eckersley will not be offered a new contract, while Jason Holt, Brad McKay and Scott Robinson will also move on at the end of the season.
“It’s important that we change the squad a bit and get better players in,” he told BBC Radio Scotland.
“We’re a Championship team that’s been promoted, so we must make sure we bring quality in.
“I’ll be looking at four, five, maybe six players.”
Neilson added: “I want to keep hold of the ones I want.
“We’ve had a great season, but, when we go up, it’s going to be a whole new ball game.”