Christian Gytkjaer scored the winning goal in the second half for Monza’s first win of the season to move off the bottom of the table.
Juventus had Angel di Maria sent off in the first half, the 10th dismissal of the Argentine’s career.
Monza were promoted to Serie A for the first time last season.
The club, owned by former Italy Prime Minister Silvio Berlusconi, missed out on automatic promotion but emerged through the play-offs.
Centre-back Pablo Mari is on loan from Arsenal, with Italy internationals Matteo Pessina and Stefano Sensi also in their side.
Samuele Birindelli, the son of former Juve defender Alessandro Birindelli, came on for Monza with five minutes remaining.
Juventus, meanwhile, have had a poor start to the season and manager Allegri’s position is under threat.
The side lie eighth in Serie A, having won just twice in seven games, and have also lost both of their Champions League games so far.
