La Liga and European champions Madrid won 2-0 on Wednesday night to complete a 5-1 aggregate victory.
Pique, 30, who has been at Barca for nine years, said: "We are not in the best moment, as a team or a club.
"This a long process and there is room for improvement, We must stay as close as possible and keep moving forwards."
Real were without Cristiano Ronaldo, who is serving a five-match suspension after being sent off in the first leg, and left Gareth Bale and Isco on the bench.
Barcelona, who are lacking in attacking options since the world record transfer of Neymar to Paris St-Germain, scored their only goal over the two legs from the penalty spot.
General manager at the Nou Camp, Pep Segura, claims the signings of Philippe Coutinho and Ousmane Dembele are "close".
Liverpool are adamant that Brazilian playmaker Coutinho, who has submitted a transfer request, will not be sold in this transfer window but Segura has told Spanish television: "We are discussing their deals but do not know when it will be done."
Borussia Dortmund suspended Dembele after the 20-year-old forward failed to turn up for training.
"We know we have to strengthen this team and that is what we are doing," Segura added. "We have to help the team."

In the annual presidential speech to Congress, Mr Humala said he aimed to cut Peru's poverty to 15% by the end of his term in 2016.
He said his government had not yet achieved all it set out to do.
The address comes days after Mr Humala reshuffled his cabinet amid unrest over a controversial mining project.
Mr Humala took the oath of office on 28 July 2011 vowing to eradicate poverty and social exclusion.
But many Peruvians complain that he has not done enough to share the wealth from the country's exploitation of its natural resources.
"I stand at the core of my proposal," Mr Humala told Congress.
"We have begun to lay the groundwork for the great transformation that most citizens of our country crave: inclusive growth... although we have not achieved everything we set out to achieve."
He told Congress that "all beginnings are tough", but vowed to extend social programmes to lift more people out of poverty.
Mr Humala's first 12 months in office have been marked by disputes and conflict and his approval rating fell to a new low of some 40% this month.
Earlier in July, five people were killed in clashes with police during protests against a huge mining project in the Cajamarca region of northern Peru.
Last Monday, Mr Humala responded to the public anger at this and other social and environmental controversies by reshuffling his cabinet for the third time in his term.
He told Congress the government was "aware of the persistence of social discontent and unsatisfied hopes among a sector of the population that wants a better quality of life".
But said that the country needed to "overcome this culture of conflict".
Barcelona defender Gerard Pique says he felt "inferior" to Real Madrid for the first time after being thrashed by their rivals in the Spanish Super Cup.
[NEXT_CONCEPT]
Peru's President Ollanta Humala has marked his first year in office by pledging to increase social spending to help the country's poorest people.