Inter had collected just one point in their first three league games under the returning head coach, but goals from Mateo Kovacic and Andrea Ranocchia ensured a much-needed win at Stadio Marc' Antonio Bentegodi.
Kovacic struck his third Serie A goal of the season after 19 minutes with a cool finish following a Mauro Icardi knock down.
Just before the hour, Ranocchia converted a Danilo D'Ambrosio cross to double Inter's advantage, before Chievo substitute Ruben Botta rounded off a miserable evening for the hosts by getting sent off just eight minutes after coming on against his parent club.
Mancini's men pushed for more goals in the closing stages but were unable to stretch their lead, with Fredy Guarin particularly wasteful in the final third.
Inter move up to 11th thanks to their first Serie A win since October, while Chievo remain just above the relegation zone.
Inter controlled the early exchanges and could have twice taken the lead as Kovacic and Rodrigo Palacio both had sights of goal, but resolute Chievo defending kept them at bay.
The visitors nearly paid for their profligacy soon after, as they failed to deal with a Chievo free-kick properly and allowed Mariano Izco a free shot on goal.
Izco scuffed his effort from the edge of the penalty, but the ball looped up for Alberto Paloschi and the outstanding Samir Handanovic had to make a point-blank save to keep the forward out.
Inter continued to dominate possession despite Chievo's brief resurgence, and they eventually broke the deadlock through Kovacic after 19 minutes.
Yuto Nagatomo delivered a dangerous cross to the back post for Icardi, he headed down to Kovacic and the Croatia international calmly side-footed the ball beyond Albano Bizzarri from six yards.
Chievo came close to levelling 10 minutes before the break when Riccardo Meggiorini took advantage of more slack Inter defending, but his acrobatic volley was saved low down by Handanovic.
The opening minutes of the second half saw both sides go close, but Nagatomo screwed his shot wide and Paloschi wasted a swift Chievo counter-attack with a poor finish.
But Inter put daylight between themselves and the hosts 10 minutes after the restart, as Ranocchia met a low cross from D'Ambrosio and guided the ball past Bizzarri.
Inter had further chances to add to their tally immediately after the second goal, but Icardi was unable to convert a low cross and Guarin saw a long-range free-kick easily saved.
Any hopes Chievo had of getting back into the game were dealt a fatal blow with 18 minutes remaining when substitute Botta was dismissed.
The on-loan Inter midfielder received two bookings in quick succession after leading with his arm in a challenge with Zdravko Kuzmanovic before being immediately cautioned again for apparent dissent.
Bizzarri kept the score respectable in the closing minutes with good saves from Guarin and Inter substitute Pablo Daniel Osvaldo, but it mattered little as Mancini's side earned their new boss a much-needed three points.
