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A Pierre-Emerick Aubameyang brace and goals from Ronald Araujo and Ferran Torres sealed a memorable win for the Blaugrana, who had lost its previous five games against Carlo Ancelotti's league leader.
The win meant Xavi became just the third coach to win his first Clasico match by four or more goals in LaLiga history, after Ferdinand Daucik in 1951 and Helenio Herrera in 1959.
Barca was superior to its host in every department and Xavi believes his side can be a match for anyone if they continue to perform at that standard.
"We didn't expect to be so superior at their stadium," he said.
"They've had a rest week and we arrived from Turkey [after playing Galatasaray] on Friday. We played a spectacular game and I leave very satisfied.
"We have been far superior to Madrid, in the game and the result. We leave with tremendous satisfaction and a smile from ear to ear.
"I'm very happy for Barcelona for what it represents. It can change the dynamics of the present and future.
"Has Barca returned? I'll leave it to you. But yes, this is the idea and the game model to follow. We have very good players and they enjoy themselves on the pitch.
"And I would highlight how they work, how they help the team. We are a family in the locker room and is very important.
"I'm very proud and happy. Barcelona fans have to enjoy it because lately we haven't had much joy, especially in the Clasicos. Playing like this we can compete with anyone.
"I'm very happy for the players, they've shown they're great footballers and it means a lot. I'm a rookie as a coach and winning 4-0 here strengthens me a lot. Barca has to compete playing this way."
Despite the result, third-placed Barca are still 12 points adrift of leaders Madrid, although they do have a game in hand.
Xavi stopped short of claiming Barca are back in the title race, but he did say there is still plenty left for his side to achieve this season.
"There is work to do," he said. "We won three points, it's not a title. Let's keep working, there's a lot left. The objectives have not been covered, not even for this season.
"I've already said it, as well as being a coach, I feel for this club. I think it's the best club in the world. We're going to celebrate now and enjoy it."
Barca is next in action on 4 April when it hosts second-placed Sevilla.