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.

The 27-year-old still had two years left on his existing deal but he has now committed himself to New Road for another season.
He is the ninth player in five months to sign a new deal with Worcestershire, who will be back playing in County Championship Division Two in 2016.
"It was very pleasing to see the other contract extensions," said Shantry.
"It shows the level of morale at the club, even though we had a difficult year in Championship cricket. We've got a group of players here that are only going to improve."
Shantry, Ed Barnard, Joe Clarke, Brett D'Oliveira, Tom Fell, Joe Leach and Charlie Morris have all signed for three years, England Lions man Ross Whiteley has signed for four years and Tom Kohler-Cadmore has agreed a two-year deal.
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.
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Worcestershire left-armer Jack Shantry has signed a contract extension with the relegated county until 2018.