Both of Aguero's boots made contact with the upper part of Luiz's left leg as the two tangled late in City's 2-1 win, which secured a place against Wigan in next month's final.
"I've contacted @DavidLuiz_4 and apologized for what happened during the match," Aguero wrote on his official Twitter account, @aguerosergiokun.
"It was an impulse reaction that shouldn't have happened."
The Argentina international will face no action over the challenge after the Football Association confirmed referee Chris Foy had seen at least part of the incident and acted on it during the match, meaning the governing body were unable to deal with it retrospectively.
Luiz suffered no serious harm in the challenge and played on until the end of the match - but in the aftermath did call on his fellow South American to apologise.
Luiz said: "Aguero is a great player but he needs to be honest with himself and say, 'I don't like to do bad tackles'.
"Kids see the game and they want to see the good things and not the bad - I would apologise. Football is beautiful, but sometimes in games people have five seconds like that - I forgive him because I have a good heart. I don't like to beat people.
"Sometimes in a game people have five seconds when they are angry and do things they don't like to do. I think it was that.
"I never saw Aguero do a bad tackle on other players."
- Sports & Recreation