Neymar and Alexis Sanchez scored the goals for the hosts, before a last-gasp consolation by Jese Rodriguez wrapped up the result.
And Xavi, although being happy with the result and his side's Liga standing, called for an improvement in order to remain at the pinnacle in Spain.
"We were the better side and for that reason we are quite happy, but we have to improve a lot of things," the 33-year-old told reporters.
"This can be done through playing, and now we are six points ahead of Real Madrid we can be quite relaxed about it.
"Real Madrid are a side with excellent players. But we were in charge, and Neymar's goal gave us a lot of confidence. In a Clasico, going into the lead is 70 per cent of the battle."
The veteran midfielder also commented on complaints by Madrid players who claimed a challenge by Javier Mascherano towards Cristiano Ronaldo should have been awarded with a penalty for los Blancos.
"You can say that it was either a legal charge or an illegal one. The referee did not blow for it and so it was all the better for us," he added.
Barcelona stay at the top of La Liga, with Atletico Madrid trailing them by a point thanks to their 5-0 trashing of Betis on Sunday.