Bielsa confronted the home bench after they attempted to slow down the clock with their team 1-0 up and minutes away from inflicting on Leeds their first league defeat of the season.
As it turned out, the Whites’ unbeaten record remained intact thanks to Jack Harrison’s 89th-minute equaliser, which actually put the visitors a point clear at the top of the Championship.
Bielsa said: “I take the responsibility for this situation because I am in football for longer than my colleague.
“I have to understand that circumstances that you have during the game, you don’t have to take them as they are.
“If you win or you lose, it means a lot for us and sometimes you behave in a kind of way and just have to lament how we behaved.
“I think I have the obligation of not behaving like this and I shouldn’t allow myself to behave like this because when you get more experienced you should moderate your behaviour. That’s why I apologise.”
Opposite number Neil Harris revealed the two parties have already put the incident behind them and was honest as to what caused it in the first place.
Harris said: “We should have given the ball back quicker in our dugout but we don’t because you had two minutes to go and we’re 1-0 up at home and obviously they got excited when they scored a goal.
“But I’ve spoken to Marcelo downstairs and there are no problems whatsoever, it’s fine.”
Millwall, who beat Leeds in this fixture when they were top of the table last season, went ahead in the 55th minute when Jed Wallace struck at the back post after Jake Cooper flicked on Ryan Leonard’s long throw.
Leeds grabbed the equaliser when Harrison found the bottom corner with a fine strike from just outside the area.
Millwall could still have snatched it, however, but Tom Elliott sent a header from Wallace’s cross against the post in the second minute of stoppage time.
Bielsa, who was without Kemar Roofe and Pablo Hernandez at The Den, said: “We draw the game at the end, we lacked the stability to recover for the last five minutes.
“For us, it’s still hard to resolve situations where we know in advance that we’re going to face them.
“We know we’ll have some kind of things during the game and we prepare to find solutions to that and it’s hard to avoid this.
“It means that we still have to improve - if you have a look for their corners, the set-pieces, the ball made a high movement and was going down, where usually the ball arrives to the box with a lot of strength.”
Following what was an improved performance from his side, Harris added: “I think it was a game of not a huge amount of chances and I thought we matched Leeds throughout the course of the 90 minutes.
“When you’re playing against a top side, and Leeds are a top side, they’re going to create the odd moment of brilliance and they had a couple of chances with clever play, or slight errors by us.
“But we scored a perfectly good goal, created other chances and, in the end, should have won the game. Am I sitting here bemoaning only getting a point? No.
“I could be negative about conceding a last-minute goal, which is disappointing from us, but I’d rather concentrate on us being really pleased with a point.
“It was a performance I knew was coming because the group’s attitude in the last couple of weeks has been brilliant.”