Madrid-based newspaper AS printed a front page on Monday pointing out that Real had picked up more points than Barca in the final 20 games of the season, since Zinedine Zidane became coach.
"In Madrid they can keep inventing stories, and we'll keep winning trophies to add to our museum," Pique told a news conference on Wednesday when asked about the AS front page.
"The difference has been the level of talent of our players and their competitiveness. In the last few years we've had so much talent here, homegrown and bought in. We have to enjoy what we are and what we have won," he said.
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"Real won a lot in the past and they may start winning again in the future, but now it's our turn."
Pique also said there was a huge level of satisfaction in the team at winning the league, despite Barca's dip in form towards the end of the campaign and failing to reach the Champions League final, where Real will play Atletico Madrid.
"Winning the league was the best moment of the season because it’s one of the most competitive and toughest leagues around. We had to give our very best to win it," Pique said.
"We are a winning team and the fans get used to winning. Some say winning so often loses its value, but I think the players are as excited to have won this league as any other. We love winning. Winning six out of eight titles is a brutal record."
Barca can complete the double next Sunday by winning the King's Cup against Sevilla, while Real could win an 11th European Cup the following week.
Pique denied that Real winning the Champions League would overshadow Barca’s achievements.
"No. A double is a double, and if Real win the Champions League we'll have to congratulate them," he added.
"Everyone follows their own path. But statistics are there for everyone to see."
(The story is refiled to remove Barcelona from byline)
(Editing by Hugh Lawson)
