Mourinho, who won three Premier League titles in two spells at Chelsea, gestured with three fingers towards the home crowd following the final whistle of Manchester United’s 2-2 draw at Stamford Bridge.
The United manager had earlier reacted angrily to the celebrations of Chelsea assistant coach Marco Ianni after Ross Barkley’s 96th-minute equaliser.
Sarri agreed that Mourinho’s past achievements at Stamford Bridge should not be forgotten by Chelsea fans.
Sarri said: “Yes. Mourinho here has won and so I think that we need to respect him and to respect everybody who has won a trophy for the club.”
Mourinho also won one FA Cup, three League Cups and a Community Shield in west London, but has struggled in the away dugout at Stamford Bridge, losing on his three previous visits with Manchester United.
Minutes away from an impressive win on Saturday, thanks to an Anthony Martial double, Barkley’s late finish prompted exuberant celebrations from Ianni, for which Sarri apologised.
Speaking immediately after the game, Sarri said: “I am not really happy. After I spoke to Jose, I understood we were on the wrong side of the situation.”
Mourinho accepted the apology from Sarri and his assistant, but did not speak about his relationship with Chelsea’s fans.