The visitors came close to scoring in the opening stages when Robinho curled an effort inches over the bar.
Cagliari looked to pose a threat as Nicola Murru tried his chances with a long-range effort that sailed wide.
Moments later Milan should have found the back of the net. Keisuke Honda was slid through on goal by Balotelli, however, his first-time strike was pushed behind by the keeper.
In the 28th minute, the home side took the lead. A poor clearance from Marco Amelia was collected by Mauricio Pinilla, who then picked out Sau in the area. The striker took a touch, before rounding the keeper and sliding the ball into the net.
Avramov was proving hard to beat in the Cagliari goal as Honda saw the keeper palm away his close-range header.