Luis Suarez made an instant impact on his Barca debut, laying on Neymar's fourth-minute opener but Pique gave Real a route back into the contest when he handled Marcello's cut-back.
Cristiano Ronaldo made no mistake from the 35th minute spot-kick and the European Champions overwhelmed La Liga’s leaders after the break.
"In the first half we were the better team," Pique told reporters. "Neymar scored, [Lionel] Messi had a great chance, but then came the penalty.
"It would be nice to know what the criteria is because it's true the ball touches my hand but I couldn't get it out [of] the way.
"He kicked the ball and I can't disappear. The ball hit my hand and the whistle [was] blown then.
"The referee didn't say anything. I threw myself to the ground to use my body and I couldn't put my hands behind my back."
Ronaldo's equaliser was the first time Barcelona's defence had been breached in La Liga this season, but Pique found himself on the floor at the wrong moment once more as Pepe planted home a 50th-minute header.
A Real corner led to that second goal and the hosts capitalised from a Barcelona set piece with a lightning counter-attack from which Karim Benzema sealed the points.
"In the second half we suffered because they got the second goal and used very fast counter-attacks. Madrid's second goal changed the game," Pique added.
"We were really good in the first half. After that we suffered, but it’s the Bernabeu, where we’ve often suffered, but we didn’t do badly."
