"One match for Rodrigo Paiva? You should be ashamed for that delinquent dressed in a suit," Pinilla wrote on his Twitter account before comparing the ban to the four-month suspension given to Uruguay's Luis Suarez for biting Italy's Giorgio Chiellini last week.
"Suarez for a bite risks his career...A press chief gets one game for a punch in the face? FIFA??
"I demand that FIFA hand Rodrigo Paiva an exemplary punishment equal to that of my colleague Suarez...!!! This is even more serious! They have the images," he added.
FIFA's disciplinary committee was also investigating the incident, meaning Paiva could face further sanctions.
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In a statement on the website of the Brazilian Football Confederatin (CBF), Paiva said: "I respect, as I always have, decisions taken by FIFA. The case is being investigated by FIFA's disciplinary committee, and they already have proof of reprehensible conduct by the Chilean delegation that will show the truth of what happened."
Chile's media spokesman Maria Jose said his Brazilian counterpart "hit" Pinilla.
Paiva told Brazilian media there was shoving by both sides as the two teams left the field.
"I was not the only one involved. Pinilla came looking for me and I just defended myself," he was quoted as saying.
"I just pushed him," he added.
"They started to insult us, there was some shoving but no assault.