Mourinho celebrated wildly by jumping into the crowd and embracing fans after a defensive mix-up allowed Fernando Torres to secure a 2-1 victory in the Premier League.
"I didn't shake hands. I didn't want to," Pellegrini told reporters at Stamford Bridge after his side suffered their third away league defeat of the season.
Mourinho apologised for his loss of decorum and said he was trying to find his son when he leapt into the crowd.
"If they feel I did something wrong I apologise," said the Portuguese. "This is the drama of the last minute.
"He (Pellegrini) lost the game in the last minute and I have sympathy with that. In other countries and cultures we shake hands before the game and that is it.
"If he was sad because of the result I understand. If he was sad because I did something wrong I accept that too."
Mourinho and Pellegrini have been rivals since the Portuguese replaced the Chilean at Real Madrid in 2010 following a second-place La Liga finish.
"Second place is just the first loser," Mourinho said before a match against Pellegrini's new club Malaga
"If Madrid were to fire me I wouldn't go to Malaga. I'd go to a top-level team in Italy or England."
The victory moved Chelsea up to second in the table, level on points with Liverpool in third and two points behind leaders Arsenal.
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