Inter's week spent blaming officials' decisions for their disappointing campaign ended with another controversial penalty call as Pinilla fired the rehoused Sardinians ahead midway through the second half before hitting the winner 15 minutes from time.
Cagliari have left the doomed Stadio Is Arenas to move back to the Rocco, where they closed out last season, but the 600-mile journey north east did not adversely affect their players, who now find themselves in the top half of the Serie A table.
Inter, meanwhile, drop to seventh on the back of a second consecutive defeat, with the prospect of Champions League, or even Europa League football for next season fading fast.
Esteban Cambiasso cleared Daniele Conti's fierce finish before Davide Astori swept up Javier Zanetti's treacherous cross in the early exchanges.
Then, minutes after Samir Handanovic had denied Daniele Dessena, Inter raced up the other end.
A deft exchange between Cambiasso, Tommaso Rocchi and Ricky Alvarez saw the veteran midfielder clip a post with a deft flick.
Alvarez would dominate the next 10 minutes, his surge on goal closed down by Gabriele Perico before Luca Rossettini blocked his ball across the box for Rocchi.
The former Velez Sarsfield man then wowed the Rocco as he nutmegged Perico with a backheel only for his finish to sail wide.
Inter would waste two more chances to take the lead before the break.
The build-up from a Nerazzurri corner saw Mateo Kovacic blast over the bar with Alvarez darting a powerful left-footed drive just over the crossbar.
Rocchi betrayed his 35 years to threaten an opener in acrobatic style following the restart, but the former Lazio man's shot on the turn was parried by Agazzi.
It was a moment Inter would live to regret as Cagliari took the lead from the spot in the 61st minute.
When Pinilla went down easily in the box from a Matias Silvestre challenge, the latter was booked before the Chilean dusted himself off to beat Handanovic down the middle.
Inter came close to an equaliser when Rocchi tried to convert Silvestre's spilled header, and Fredy Guarin had needed just a little more accuracy with a daisy-cutter.
Cagliari's second goal may have lacked the grace of their opponents' keen efforts but Pinilla could not be accused of gamesmanship on this occasion.
A swift exchange with Santiago Cabrera brought the Chilean stumbling into the area, where he somehow recovered to dig his sweet finish high into the roof of the net.
Inter were now at sixes and sevens and were only spared the embarrassment of conceding a third goal by the width of the bar as Ibarbo raced onto a Thiago Ribeiro ball over the top.
Cagliari's work was done, however, and the travelling Sardinians rode out a comfortable finish to reach 10th place.