Ex-Manchester City manager Mancini, 51, claims Sarri used homophobic language towards him during a touchline row when their teams met in the Coppa Italia.
However, the suspension was not for homophobia, for which the punishment could have been greater.
Sarri, 57, has also been fined 20,000 euros (£15,000).
Napoli were beaten 2-0 in Tuesday's quarter-final, so Sarri will not serve his ban until next season.
Mancini, in his second spell at Inter, was also fined 5,000 euros (£4,000) after he was sent off towards the end of the match.
Warning - the section below contains language some readers may find offensive
Mancini said Sarri shouted "poof" and made another homophobic slur in an altercation following Inter's second goal.
Sarri claimed not to remember his exact words, but acknowledged he had been "fired up and angry".
He added: "I was not discriminating against anyone. If I did indeed use those words, then I apologise to the gay community."
Italian snowboarder Arianna Cau said Sarri's ban is too short.
"I am not a judge, so I can't really say what the punishment should be, but it should be more serious," Cau told BBC World Service.
"People understand that in Italy we have a problem. If we don't have a serious punishment, then we will never fight this war.
"They say homophobia doesn't exist in Italy, but that's not true. The moment has arrived to start taking this seriously."
Napoli manager Maurizio Sarri has been banned for two Italian Cup matches for "highly offensive" comments made to Inter Milan boss Roberto Mancini.