Montolivo decided a game affected by rainy conditions in the 25th minute as the Rossoneri responded to Juventus' 2-1 win at Inter Milan to keep their own bid for the Scudetto alive.
Chievo's woes were compounded by a red card for Dario Dainelli as they ended the day nine points above the drop zone.
The rain, which never stopped falling throughout the 90 minutes, had already started to form puddles when the teams took to the field at the Stadio Marc'Antonio Bentegodi.
The stage was set for a scrap and Milan rose to the challenge to keep their slender title hopes alive.
The first chance fell to Chievo in the 15th minute when Cyril Thereau's cross was diverted towards his own goal by Philippe Mexes, but Christian Abbiati was able to keep the ball out with Daniele Bonera clearing his lines.
Thereau then tested Abbiati with a shot which lacked the pace to cause the Milan goalkeeper any problems.
Milan took the lead with practically their first chance of the game.
Mario Balotelli's free-kick was only parried by Christian Puggioni and Montolivo pounced on the loose ball to put the visitors in front.
Another free-kick from Balotelli was spilled by Puggioni, who this time recovered the ball at the second time of asking.
Balotelli renewed his acquaintance with Puggioni three more times from set-pieces outside the area, but the Chievo keeper came out on top each time.
At the other end Luciano had the best chance to bring Chievo level, but his effort was turned over the crossbar by Abbiati.
Dainelli then received his marching orders for a foul on Balotelli on the edge of the penalty area which earned him a second yellow card, although no further punishment arrived from the free-kick as Balotelli was again denied by Puggioni.
One final shot from Balotelli was saved by Puggioni, but a second goal was not required for Milan to move within 11 points of league leaders Juventus.