The Chilean striker came off the bench to net two second-half goals and send Andrea Stramaccioni's inconsistent side to its second consecutive Serie A defeat.
The Nerazzurri were dominant for much of the game, but as they failed to take their chances in front of goal, a second-half double from Mauricio Pinilla condemned them to their fourth defeat in five games.
The hosts were forced to withdraw Andrea Cossu after just seven minutes due to injury, and Inter's patient start to the game enabled the visitors to control the ball during the opening stages.
A tame shot from Daniele Dessena early on was the only goal-mouth action until a sudden attacking injection saw Esteban Cambiasso and Tommaso Rocchi carve apart the Cagliari defense. After some sublime interplay, the Argentine looked to set to round off the move with a delightful chip but was denied by the post.
Rocchi and Ricardo Alvarez continued to work hard in the final third, but neither could find a decisive touch with the latter of the pair seeing his shot blocked after a coruscating dribble into the box midway through the half.
Center backs Davide Astori and Juan Jesus both saw their attempts on goal drift harmlessly wide at either end before Victor Ibarbo missed a golden opportunity in the 35th minute.
The Colombian looked a certainty to convert from Nicola Murru's cross, but Alvaro Pereira remained alert and managed to divert the effort from six yards behind.
The Nerazzurri continued to dictate the tempo of the game after the break, and Michael Agazzi was called into action twice soon after the restart. Having stayed tall to keep Rocchi's powerful effort out at the near post, he then got a hand to Fredy Guarin's shot to send it behind.
Cagliari was content to play on the counterattack, and after having two penalty claims rejected, the referee pointed to the spot in the 63rd minute when Matias Silverstre felled Pinilla. The Chilean stepped up and rifled the ball straight down the middle to put Ivo Pulga's men ahead.
It could have been two six minutes later had it not been for a lack of direction on Thiago Ribeiro's header, which enabled Handanovic to make a comfortable save.
Inter continued to prove profligate in front of goal, and its fate was sealed in the 76th minute. Once more it was Pinilla with the goal, stabbing the ball into the roof of the net after a delightful give-and-go with Matias Cabrera.
Ibarbo almost compounded the misery of the Nerazzurri faithful but he could not keep his late effort down and watched it clip the crossbar and go behind.
Following the defeat, Inter could now find itself 11 points off of third place should rival AC Milan overcome Napoli later on Sunday. Cagliari, meanwhile, moved into the top half of the table with the win.