Barca eased the pressure on head coach Ernesto Valverde with their victory at Camp Nou, moving onto 10 points from six La Liga games.
While not wanting to make excuses, Pique felt Barcelona's pre-season, which saw them travel to Japan and the United States, has impacted their start to the campaign.
Lionel Messi made just his first start of the season after recovering from a calf injury, only to come off at half-time due to an adductor problem.
"The pre-season did not help at all to start well in the competition, a lot of travel and little training," Pique said post-match.
"Many of us notice it and we don't have the tone yet, not only Messi, but we must continue because we are forced to win, it is a setback.
"Look now what has happened to Messi, but there are no excuses, in the team there are qualified people to replace him and move on, although we hope he will be out as short as possible.
"There are quite a few players who have not achieved the tone because it is what it is. I do not know the severity of his [Messi's] injury and I hope it is as little as possible.
"There are no excuses. Although it is the best there are no excuses in Barcelona, although the pre-season was not the best."
Goals from Antoine Griezmann and Arthur helped Barca past Villarreal, who pulled one back through Santi Cazorla.
Pique said the three points were vital for Valverde's team as they now sit a point behind league leaders Granada.
"It was important to win so as not to disengage from the top teams. With both goals and Leo's injury, the team has pulled back a bit," he said.
"They have advanced the lines and marked us. In the second half, it was difficult."
Barcelona make the trip to Getafe in their next outing on Saturday with Pique aware the Catalan faithful will expect nothing less than another win.
"I understand the people, they pay the entrance fee and they deserve to express what they feel," he told reporters, via Marca .
"We would like to win all the games, but the players are people and emotionally and personally we have our life and it is difficult to yield 100 per cent and we are not machines.
"We try to do our best. I wouldn't give it more importance. We have to focus on Getafe."