Stephan El Shaarawy — who had scored five goals in his past four matches — then found a way through but, under pressure from Nagatomo, he couldn't get a clean shot on goal and fired wide.
Montolivo had the ball in the back of the net in the 40th minute with a 30-yard screamer into an empty net but the referee ruled it out for a foul moments earlier by Urby Emanuelson on Handanovic as the Inter goalkeeper came flying out to punch the ball clear.
Milan was buoyed minutes into the second period when Nagatomo received a second yellow for handball.
Bojan went closest for Milan as Handanovic made a rare mistake and spilled a shot but the former Roma striker slipped as he raced towards the ball, with only the Inter goalkeeper to beat.
Elswehere, Napoli remained joint top with Juventus after beating Napoli 2-1. Marek Hamsik and Goran Pandev scored in the first half, either side of Giampiero Pinzi's leveler to set up a mouthwatering contest between the top two after the international break.
Earlier, defending champion Juventus stretched its Serie A unbeaten streak to 46 games with a 2-1 victory at Siena.
Claudio Marchisio scored what proved to be the winner for the league leader six minutes from time. Andrea Pirlo gave Juventus the lead with a trademark free kick in the 13th minute but Emanuele Calaio leveled in first-half stoppage time.
Also Sunday, high-flying Lazio cruised to a 3-0 win at struggling Pescara. Hernanes opened the scoring with a stunning 30-yard free kick in the fifth minute, before Miroslav Klose grabbed a brace.
In the lunchtime fixture, American Michael Bradley scored his first goal for Roma in a 2-0 win against Atalanta. Erik Lamela scored the opener.
Stevan Jovetic continued his impressive start to the season as he scored his fifth goal of the season to send Fiorentina on its way to a 1-0 victory against Bologna.
Catania also got off to a good start in its 2-0 win against 10-man Parma as Alejandro Gomez gave it the lead with less than two minutes on the clock.
Parma defender Yohan Benalouane was sent off midway through the second half for a second bookable offense, before Gonzalo Bergessio sealed the result.
Elsewhere, Cagliari's new coaches Ivo Pulga and Diego Lopez got their reign off to a perfect start, recording the Sardinian team's first victory of the season with a 1-0 win at Torino.
Nene scored the only goal of the game in the 74th minute from the penalty spot following Torino defender Kamil Glik's handball.