The former Manchester City and Galatasaray boss replaced Walter Mazzarri last month but had failed to win in his opening three league games back in charge at San Siro.
However, goals from Mateo Kovacic and Andrea Ranocchia ended Mancini's wait for three points as they powered past struggling Chievo at Stadio Marc' Antonio Bentegodi.
Kovacic slotted home from six yards after 19 minutes following Mauro Icardi's knock-down to break the deadlock, and Ranocchia doubled Inter's lead just before the hour.
Chievo were frustrated by outstanding Inter keeper Samir Handanovic on their rare attacks, and their evening ended in further disappointment when substitute Ruben Botta was sent off eight minutes after coming on.
The Inter loanee was given two bookings for first leading with an elbow on Zdravko Kuzmanovic, and then appearing to react to the referee's decision with dissent.
In Monday's other Serie A fixture, Empoli were held to a goalless draw by a resolute Torino defence.
The hosts had the better chances at Stadio Carlo Castellani with Matias Vecino and Simone Verdi both going close, while Fabio Quagliarella threatened for Torino with an audacious overhead kick.
Torino finished the match with 10 men after Pontus Jansson was sent off in stoppage time for a second bookable offence, the defender penalised for lunging in on Piotr Zielinski.
