The Italy midfielder kept the club's hopes of finishing second in Serie A alive as Massimiliano Allegri's side ended Chievo's run of three straight wins in a sodden Verona.
The Italy midfielder took advantage of a goalkeeper error to stab home, and despite Mario Balotelli doing everything in his power to make the points safe, the Rossoneri had to settle for a single goal.
Center back Dario Dainelli saw red for the home side late on and his dismissal put to bed any chance of Chievo getting back into the match.
After a slow start, Milan almost gifted the home side the opening goal as Cyril Thereau's cross from the right was spilled by Christian Abbiati. The ball bounced goalwards off Daniele Bonera, but the goalkeeper recovered well and smothered the ball just short of the line.
Despite running out winners in all of their four previous Serie A matches, Milan were struggling to put their stamp on the game, and Abbiati was called into action again when Thereau's week effort almost sneaked in at his near post.
The Stadio Marc Antonio Bentegodi pitch was incredibly wet, with the ball regularly skidding off the surface, and Massimiliano Allegri's side used this to its advantage to break the deadlock.
Balotelli's low free kick bounced just in front of goalkeeper Christian Puggioni, and after he could only parry it back into the six-yard box, Montolivo was on hand to bury the rebound.
Former Manchester City striker Balotelli had sensed a weakness, and again tried his luck with a dead ball. Puggioni failed to gather the ball at the first attempt, but this time managed to recover before any of the onrushing Milan players could double the lead.
The match was slowly turning into a scrappy encounter punctured with a Balotelli free kick, and that pattern continued as the Italy international tried his luck once more, with Puggioni, who had obviously been affected by the previous two dead balls, nervously punching the ball away.
Balotelli's set pieces continued to worry the Chievo goalkeeper as the halftime break drew nearer, with Puggioni opting to punch a fourth effort of the half when catching seemed the sensible option.
If Puggioni thought the second half would bring him some respite from the bombardment of free kicks, he was incorrect as Balotelli again tested him with a curling effort. The goalkeeper again chose to parry the shot, but this time at least turned it behind.
The visiting side was relentless in its quest for a second goal, and after Sulley Muntari fired just wide from distance, Balotelli again tested Puggioni from a set piece.
The goalkeeper was called into action once more, this time from open play, as he got down well to block Robinho's effort, and all his good work almost proved crucial as Chievo went close at the other end.
Substitute Luciano cut onto his left foot on the edge of the area and fired in a powerful shot, only to see Abbiati pull off an unbelievable save to deny him an equalizer.
The home side was coming more into the game, but the chances of taking any points from the match were dealt a huge blow as Danielli was dismissed for a second bookable offense. Balotelli naturally tested Puggioni from the resulting free kick.
With only 10 men Chievo tried to push forward in the hope of claiming a draw, but the club failed to test Abbiati as the San Siro outfit held on.