Xavi expects taking "three golden points" off Atletico Madrid to give Barcelona "tremendous morale and confidence" in the LaLiga title race.
Ousmane Dembele scored the only goal of the game at the Civitas Metropolitano on Sunday to put Barca three points clear of Real Madrid at the top of the table.
The Blaugrana defended resolutely to take full advantage of Madrid's 2-1 defeat at Villarreal on Saturday, Ronald Araujo clearing off the line to deny Antoine Griezmann a stoppage-time equaliser against his former club.
Stefan Savic and Ferran Torres were sent off for a late off-the-ball clash, but that did not take the gloss off a big win for Barca in the absence of suspended striker Roberto Lewandowski.
Barca had been held to a frustrating draw by Espanyol in their first LaLiga game after the World Cup, so Xavi knew the importance of beating Diego Simeone's side in the capital.
He said: "It's a victory, but it's not just three points, it gives us tremendous morale and confidence for the remainder of the league.
"We are capable of winning in a field as difficult as this, suffering, being a family. Without playing as excellently as we did against Espanyol, today we have known how to suffer and we took three golden points. We hit the table to be candidates for this league [title]."
Xavi added: "Three golden points. We have taken advantage of our chance. They have pressed, and we have known how to suffer.
"And how important it is to keep a clean sheet. Just as I said on the Espanyol game that we deserved more, today perhaps without playing so well we got three points."
Captain Sergio Busquets felt the manner in which Barca dug deep to see out the win bodes well.
He said: "We started well, and we went ahead, but from then on they have pressed very well, it was difficult for us to have the ball and create chances.
"In the second half, although they came out very strongly, we managed to control the game. In the end, we suffered, but three very good points."