Sergio Ramos' header nine minutes from time proved to be the winner after Karim Benzema and Lionel Messi had exchanged goals in the first-half.
But Pique and his teammates were left aggrieved after Ramos appeared to fell Adriano inside the box with Barca keeper Victor Valdes even receiving a red card for his protests after the final whistle had been blown.
"We said to the referee that it seemed clear to us, but we already know how these things go. Against Real Madrid, if the game is close, just by chance things tend to go in their favour," said Pique.
"We are left with the feeling that a draw would have been a fair result if we had been given the penalty, but we cannot make this excuse because this team has too much quality to depend on refereeing decisions to win games."
The defeat was Barca's third defeat in four games and Pique admits they are struggling to find their best form.
"You can't maintain this level for 12 months. In the last four years we have always had moments in which our level of performance has dropped.
"However, it is clear that we have had three of four bad games and this is serious because Barcelona always have to win."
On the other hand victory came as even more of a welcome surprise for the hosts after manager Jose Mourinho made seven changes to the side that had beaten Barca 3-1 in the Copa del Rey on Tuesday with star-man Cristiano Ronaldo starting only on the bench.
However, even without their talisman they made the perfect start as Benzema tapped home Alvaro Morata's cross before Messi brought Barca level with a trademark left-footed finish 12 minutes later.
Ronaldo's introduction after the break though sparked the game into life and after Valdes had produced fine saves to beat away the Portuguese's free-kick and deny Morata when he was clean through, Ramos left the Barca keeper helpless as he rose highest to power home Luka Modric's corner.
Madrid's victory keeps their very slim hopes of retaining their title alive as it cuts Barca's lead over the champions to 13 points, but as Mourinho's team selection indicated of more immediate importance is Tuesday's make or break Champions League tie with Manchester United with the tie level at 1-1.
And Ramos believes that two victories against Barca is the perfect way to prepare for their trip to Old Trafford.
"It's clear that these victories are great for the confidence of the group because now we face a difficult match in the Champions League," he said.
"After the two Clasicos we are going to Manchester convinced that we can win, although with a lot of respect for the opponent who are a great team."
Elsewhere, Valencia missed the opportunity to move into fourth as Jose Barkero's late equaliser gave Levante a share of the spoils as the Valencia derby ended 2-2.
Vicente Iborra had headed Levante into an early lead before Jonas and Roberto Soldado replied for the hosts, but a slip by Jeremy Mathieu three minutes from time allowed Barkero in to salvage a point for Juan Ignacio Martinez's men.
At the bottom Deportivo la Coruna remain rooted to the foot of the table after they could only manage a 0-0 draw at home to Rayo Vallecano, but Athletic Bilbao eased their relegation fears with a hard-fought 1-0 win away to Osasuna thanks to Markel Susaeta's second-half strike.