Barca's Argentina forward Lionel Messi was also verbally abused and Espanyol will be reported to a Spanish government commission that seeks to combat violence, racism and xenophobia in sport, the LFP said on Tuesday.
A section of supporters at Espanyol's Cornella-El Prat stadium insulted Pique, his pop star partner Shakira and their infant son as part of generalised and sustained abuse of Barca, who won the match 2-0 despite having Jordi Alba sent off shortly after halftime.
Alba, a Spain international, was also verbally abused as he left the pitch, the LFP said.
Espanyol are likely to be hit with a fine worth tens of thousands of euros over the incidents.
Racism and abuse has long been a problem at Spanish football stadiums but the LFP is seeking to crack down on such behaviour following the death of a Deportivo La Coruna fan in fighting before their match at Atletico Madrid at the end of November.
Alves has been targeted by racists in the past and hit the headlines in April last year when he picked up a banana thrown from the crowd at Villarreal, peeled it and took a bite before flinging it away.
(Reporting by Iain Rogers, editing by Toby Davis)
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