The 52-year-old aimed an expletive-laden volley at the Chilean during the first half of his side's controversial 2-0 Barclays Premier League defeat by the new leaders on Sunday afternoon.
A contrite Pardew said: "I hear that it's been picked up a little bit, and I apologise for my comments 100 per cent.
"It was a heat of the moment thing. We just had words that we always have as managers, to a degree. I have apologised to him and fortunately for myself, he has accepted that."
Pardew's comments included a particularly offensive four-letter word, and he later admitted Pellegrini's involvement in the spat did not merit that.
He said: "No, I don't think it merited that word, if I am honest. As I said, I have apologised for that particular word, but we sort of tease each other a little bit and have words."
Pellegrini insisted he did not know what Pardew had said to him, but was untroubled by the exchange, although he did accuse the Newcastle boss of questioning every decision the officials made.
He said: "I'm not surprised because I think it is impossible to complain about every decision of the referee during the match.
"I know they were playing at home and they have an advantage in that sense, but it's impossible for the referee if you are complaining at every decision.
"That was my only problem with Alan, but nothing important."
The root of Pardew's complaint was referee Mike Jones' decision to rule out a 34th-minute Cheick Tiote equaliser for offside against Yoan Gouffran.
The Frenchman was in an offside position as the Ivory Coast international's piledriver flew past helpless keeper Joe Hart, although Pardew could not accept Mr Jones' judgment that he was interfering with play.
He said: "I have gone and seen him. I was right behind the shot, as the fourth official was, and it was a clean strike.
"It was a great moment for this football club to get a goal like that, so if you are going to chalk it off, you need to make sure that you are getting the point right.
"My point was did he think Gouffran was interfering with Joe Hart, and he said, 'He was in the six-yard box', and I get that.
"The guy was recovering from an offside position, he didn't want anything to do with the play, but he is not interfering with Joe Hart's vision. In fact, it goes on the inside of him.
"The Man City players are impairing Joe's vision, not Gouffran.
"It's a real technical matter that he's talking about here, and to chalk a goal off of that quality, that takes some doing, I think, and it did set the tone for the rest of the game.
"I think if you are going to call a guy retreating from an offside position, it has to be pretty clear that he is interfering with the goalkeeper's view.
"The linesman, even though he was on the wrong side, couldn't have been clear on that, and the referee definitely couldn't have been clear on that."
However, Pellegrini was confident that the referee, who consulted assistant Stephen Child before making his decision, had got it right.
He said: "There was a player offside when the other player kicked the ball. It was offside and was perfectly not valid at all."
City had taken an eighth-minute lead when Edin Dzeko converted Aleksandar Kolarov's left-wing cross, but they had to wait until the fifth minute of stoppage time at the end of the game for Alvaro Negredo to seal victory with his 19th goal of the season.
Pellegrini, whose side leapfrogged Arsenal into first place as a result of their win, said: "It was a very big result because it's a very difficult stadium to play here. We, Chelsea and Manchester United know how difficult it is to play Newcastle.
"They have a direct style with a lot of physical power, and the whole game, they were trying to score with crosses and shooting, and we were very concentrated in defending.
"Also, we scored two goals and we had two or three more clear chances to do it."
However, City left Tyneside fearing midfielder Samir Nasri, who was carried from the field on a stretcher after being felled by Mapou Yanga-Mbiwa's 75th-minute challenge, has suffered a serious knee injury.
Pellegrini said: "It's very serious. It was a very unfair kick and the player of Newcastle, it was directly a red card. I don't understand why he was not sent off.
"It's his medial ligament. We will see tomorrow how serious it is, but I don't think it's a light thing, I think it's a serious injury."