The 31-year-old's gesture came after the Catalan club reported him to the Royal Spanish Football Federation for referring to them as "Espanyol from Cornella" in January.
Cornella is a suburb of Barcelona, where the club's Cornella-El Prat stadium is based, and Pique's implication was that Espanyol do not deserve to be considered Barca's rivals.
"Celebrating the goal was special after everything that's happened in the last week," the Spain defender is quoted by Sky Sports News as saying after the game.
"Some of the fans here have been directing insults at me and my family for a long time. I was telling them, in general, to be quiet, but maybe I shouldn't have. I know a lot of good Espanyol fans. Maybe it should have been directed at the club or its owners for not reporting certain things.
"They are from Cornella, like I said the other day. I know it hurt them and I used sarcasm to annoy them but saying they play in Cornella is not disrespectful, it's a fact. They complain about that and report me, but they don't investigate insults towards my family. That is disrespectful. The least I could do was bring silence to the Espanyol fans."
Barca overcame Espanyol by a 2-1 aggregate scoreline in the quarter-finals of this season's Copa del Rey.