A depleted Nerazzurri side started the match brightly and should have taken the lead on 14 minutes through Esteban Cambiasso, who saw his chip come back off the crossbar.
Ricky Alvarez went close just before the break, thumping a fierce strike over the upright.
The striker got up to send Samir Handanovic the wrong way and break the deadlock with a well-taken spot kick.
Pinilla then wrapped up all three points for the home side as Inter switched off to allow the striker to control and poke home his second from close range.
Ibarbo hit the bar late on as Inter were unable to find a response to an impressive performance from the home side.