Mancini and Sarri clashed on the touchline after Adem Ljajic killed off the quarter-final tie at the San Paolo stadium in Naples in injury time to add to Stevan Jovetic's opener.
"Maurizio Sarri is a racist and men like him have no place in football," fumed Mancini to RAI TV after the game.
"He used racist words. He started insulting me and then shouted at me, calling me a poof."
Mancini said Sarri had used the words "frocio" and "finocchio", both pejorative terms meaning homosexual.
The Inter coach added: "But if he's a man then I'd be proud to be a poof. From someone like him, who's 60, I can't accept that, he should be ashamed.
"You have the right to argue, but not like that. I went to see him in the changing rooms and he apologized.
"But I told him that in England, he wouldn't be allowed to set foot on a pitch again."
Sarri brushed off the exchange as part and parcel of the game.
"I've seen and heard worse on a pitch," he said.
"I hope that in the cold light of day, Mancini will also change his point of view.
"In my words, there was no form of discrimination, I have nothing against homosexuals.
"The adrenalin of the pitch can push you to make bad jokes, but the arguments should finish at the end of the 90 minutes.
"I've apologized to Mancini, for me everything's ok."
During the match, Inter had brushed off their poor league form to storm into the Cup semi-finals.
Two defeats and just four points from their last four league matches had seen Inter lose command of Serie A and drop to third, four points behind Napoli.
But now they will face the winners of Wednesday's last quarter-final between Lazio and champions Juventus for a place in the final.
Inter's city rivals AC Milan will play third division Alessandria in the last four. The minnows won 2-1 at second tier Spezia on Monday night while Milan defeated Carpi last week by the same score.
Tuesday's match was a tight affair, in part due to Sarri leaving top scorer Gonzalo Higuain on the bench.
Despite 20 goals in 20 games this season, the Argentina striker didn't enter the fray until the 72nd minute but just two minutes later, Inter struck.
Montenegro international Jovetic, a summer signing from Manchester City, hit an unstoppable shot that left Napoli goalkeeper Pepe Reina rooted to the spot.
Two minutes from time, Napoli had Dries Mertens dismissed and in time added on, Serbian Ljajic sprinted 50 yards to score the killer second on the counter-attack.
That goal sparked the furious touchline altercation that will surely provoke serious ramifications once the Italian football league's disciplinary body looks into it.
The semi-finals will be played over two legs on January 27th and March 2nd.