Valencia spent the last couple of seasons trying to avoid relegation in the Spanish league.
After eight matches this season, it's already looking to dethrone Barcelona from the top of the standings.
Valencia reached second place with an emphatic 6-3 win at Real Betis on Sunday, moving to within four points of Barcelona, which lost its perfect record after a 1-1 draw at Atletico Madrid on Saturday.
Defending champion Real Madrid, which defeated Getafe 2-1 with a late winner by Cristiano Ronaldo on Saturday, is one point behind Valencia in third place. Atletico and Sevilla are another point behind.
Valencia, whose last league titles came in 2002 and 2004, struggled in recent seasons under the ownership of Singapore investor Peter Lim.
Under coach Marcelino Garcia, Valencia has already won five times this season, and it's coming off four straight wins while outscoring opponents 17-7.
"We are playing at a very high level collectively," Valencia striker Rodrigo said. "We deserve a lot of credit for what we have been doing so far this season. We know there's still a long way to go. We want more, we are an ambitious team."
Valencia got off to a great start at the Benito Villamarin Stadium in Seville, opening a 4-0 lead with goals by Geoffrey Kondogbia, Goncalo Guedes, Rodrigo and Santi Mina.
Betis, which started the weekend in sixth place and had won three of its last four matches, rallied with three unanswered goals after the 79th minute from Joel Campbell, Antonio Sanabria and Cristian Tello. But Valencia sealed victory in the closing minutes with goals by Simone Zaza and Andreas Pereira.
"We were close to equalizing ... but in the end it wasn't possible," said Costa Rica midfielder Campbell, whose goal on debut came after he received the ball from an erroneous back pass by Valencia midfielder Dani Parejo.
Betis forward Sergio Leon had a penalty kick saved by Valencia goalkeeper Neto when the visitors were winning 2-0.
Girona's struggles continued in the Spanish league with a 2-1 loss to Villarreal, extending the team's winless streak to six games after a promising start to its season.
Girona made its top-flight debut with a draw against Atletico Madrid and a win at Malaga, giving fans hopes of a successful season, but it hasn't been able to win since, losing four matches and drawing two.
Sunday's loss left the Catalan team at 17th in the 20-team standings, just outside the relegation zone.
"It shouldn't be a strange thing to see Girona near the bottom of the table," Girona coach Pablo Machin said. "This is the elite. We want to enjoy it, but we also will have to suffer. Our goal is to finish the season with three teams below us."
French forward Cedric Bakambu scored twice in the first 20 minutes to give Villarreal the victory at Montilivi Stadium in Girona. The hosts scored with striker Christian Stuani just before halftime.
It was the second straight win for Villarreal after three consecutive games without a victory. It moved to eighth place with 13 points, seven more than Girona.
Eibar and Deportivo La Coruna couldn't manage more than a 0-0 draw that left both teams near the bottom of the standings.
Eibar, winless in four matches and in 16th place, found the net with forward Sergi Enrich in second-half injury time but the goal was correctly called off for offside.
Deportivo, which has only two wins this season, stayed in 15th place with eight points.
Eibar, one point behind Deportivo, visits Real Madrid in the next round.
Malaga was beaten 2-0 at home by Leganes for its seventh loss in eight matches, staying at the bottom of the standings.
Leganes scored second-half goals with Gabriel and Alexander Szymanowski to earn its fourth victory and move sixth with 14 points.
Malaga, which has been outscored 18-4 this season, earned its only point from a draw against Athletic Bilbao.