The Chilean was on Inter's books a decade ago but never made a senior league appearance - however his two-goal showing served to frustrate his former employers who had failed to convert a number of chances to break the deadlock themselves.
Coming on for Daniele Dessena on 57 minutes, Pinilla won a contentious penalty five minutes later even though there did not appear to be much contact between he and defender Matias Silvestre.
Netting from 12 yards himself, Pinilla doubled his team's advantage as well as his own personal tally with a neat finish into the roof of the net 14 minutes from the end after fellow replacement Mathias Cabrera laid on a good assist from the right flank.
It is Inter's fourth defeat in their last five games, and injuries to Walter Gargano and Yuto Nagatomo came in addition to the absences of Antonio Cassano, Rodrigo Palacio, Diego Milito, Cristian Chivu, Dejan Stankovic, Luca Castellazzi, Gaby Mudingayi, Joel Obi and Ibrahima Mbaye.
The Nerazzurri started well as Esteban Cambiasso hit the post, while veteran Tomasso Rocchi - their only available striker - was denied a penalty when he was challenged by Daniele Dessena.
That made Cagliari's opener all the more controversial, as Pinilla appeared to dive under pressure from Matias Silvestre. He converted the spot-kick himself to edge the hosts ahead, and added a second as Inter over-committed in attack, finishing from close range after a sweeping counter-attack.
Victor Ibarbo could have added a late third, hitting the crossbar, with Inter well beaten in the end.