Lionel Messi – making his first appearance since fracturing his arm against Sevilla on October 20 – scored twice, but it was not enough to prevent a stunning reverse against Quique Setien's side.
A chaotic display at the back from Barca allowed Junior, Joaquin, Giovani Lo Celso and Sergio Canales to all get on the scoresheet – the first time they have conceded four at home since a 4-2 defeat to Deportivo La Coruna in April 2003.
The first two goals came during a hapless first-half performance, one which saw Gerard Pique and Clement Lenglet exposed on a number of occasions at the heart of the Barca backline.
Valverde was less than impressed with his side's defending before the interval - and has demanded an improvement when they return to action after the international break.
"We could not get back into the game at any time," he told a media conference.
"We were not good in the first half and they got a positive advantage. We paid dearly for our first-half display.
"We did not start well and we are lacking something in general. We lack freshness when we have to finish attacks and we have to improve that.
"We knew that we were not going to dominate the match and it would be a case of 'give and take' although we thought we would end up with a better result.
"But to analyse my team I think we should have responded differently."
Arturo Vidal – who was introduced at the break and scored Barca's second in the 79th minute – says they need to improve dramatically if they are to be successful this season.
"The game started badly and they took advantage of the two opportunities they had. We were ready, but we cannot keep losing home points. Teams cannot come here and score four goals."