Two first-half goals put the home side in control at San Siro, with Armando Izzo's reply on 85 minutes threatening to throw a spanner into the works until the Serbian's goal.
The 50-year-old head coach had only taken one point in his previous two home fixtures, but goals from Rodrigo Palacio, Mauro Icardi and Nemanja Vidic finally gave him all three points at San Siro.
Palacio opened the scoring early with a poacher's finish after Icardi's effort had been saved, with the latter heading in the second just before the break.
Armando Izzo got the visiting side back into the game with five minutes remaining, the defender firing home from close range, but Vidic's first goal for Inter sealed the points for the home side.
Inter's first home win since October moves the club to within two points of its opponent and fifth-placed Sampdoria.
Mancini, who gave Lukas Podolski his full debut as one of four changes, saw his side threaten inside the opening two minutes as Fredy Guarin played a through ball for Icardi.
The Argentina international escaped the attentions of Genoa's defenders on the edge of the penalty area and, although his low shot looked destined for the bottom corner, Mattia Perin did well to push it away.
Genoa was not allowing the home side to have it all its own way, though, and after seven minutes went close to breaking the deadlock, only for Samir Handanovic to deny Luca Antonelli from close range.
A frantic opening to the match was capped with a goal in the 12th minute as Palacio tapped home after compatriot Icardi's acrobatic volley had been saved by Perin.
The goal heralded a spell of dominance for the home side, which limited Genoa to rare counterattacking moves, while Palacio spurned a presentable opportunity from 12 yards.
However, Mancini's men got their second goal after 39 minutes as Icardi rose above the defense to head home Hernanes' corner from the left.
Genoa went close to halving the deficit 10 minutes into the second half as Iago Falque pounced on a loose ball 18 yards from goal, but his first-time effort was tipped wide by Handanovic.
The visiting side was having its best period of the game and another three chances went begging with Andrea Bertolacci and Antonelli both firing over, before Maxime Lestienne was denied by the crossbar.
Mancini found himself involved in the game more than he would have liked after 66 minutes as a clearance from substitute Zdravko Kuzmanovic struck him full in the face – knocking the coach to the floor on the touchline.
Podolski went close to his first Inter goal six minutes later with a free kick from the edge of the penalty area, but the German forward's set piece was deflected wide by Tomas Rincon.
Inter was denied a clean sheet in the closing minutes, though, as Izzo pounced on a loose ball after Handanovic had failed to hold onto Alberto Costa's strike.
However, Inter restored the two-goal advantage two minutes later as Vidic headed home Kuzmanovic's corner to secure the points.