Only 50.5 overs were possible at the MCG, but Australia still got wickets at regular intervals with Jackson Bird taking two for 53.
Azhar's 110-ball half-century saw him pass 1,000 Test runs in 2016.
Australia lead the three-Test series 1-0, having beaten Pakistan by 39 runs in the opener at the Gabba.
Sami Aslam was undone by Nathan Lyon after making just nine and, after Azhar and Babar Azam batted through the remainder of the morning, Azam edged the last ball before lunch from Josh Hazlewood to give Steve Smith his second catch.
Misbah-ul-Haq scored 11 off 13 balls with a four and a six before being brilliantly caught at short leg by Nic Maddinson off Bird.
Bird bowled Younus Khan off an inside edge to end a third-wicket stand of 51.
The wicket came shortly before the scheduled tea break and the weather meant that interval came early and play never resumed.
Azhar Ali's unbeaten 66 helped Pakistan to reach 142-4 against Australia on a rain-affected first day of the Boxing Day Test in Melbourne.