"It was a very good match overall, we dominated and even had chances to score more," Barcelona coach Xavi Hernandez told Movistar.
"We're going through a good period. Winning 3-0 at Sevilla is a good sign."
Barcelona, who host Viktoria Plzen on Wednesday in the opening round of Champions League group games, are two points behind Madrid, but Sevilla have just one point so far.
"The situation we find ourselves in is a bit difficult," said Sevilla midfielder Rakitic.
"I ask forgiveness for the results, but also for confidence and patience. We have full confidence in our coach (Julen Lopetegui)."
Vinicius Junior struck for the third game running as Madrid took an early lead at the Santiago Bernabeu, but Sergio Canales soon equalised with a goal against his former club.
Rodrygo secured a 2-1 win for Carlo Ancelotti's side just past the hour with his first goal of the campaign, ending Betis' perfect start.
"He's a special forward because he can play in any position. He's very quick and very intelligent without the ball," Ancelotti said of Rodrygo, now in his fourth season with Real.
"The apprenticeship is over and now he's a Real Madrid player in every way."
Next up for the European champions is a trip to Celtic in the Champions League.
"The most important game of the first part of the season is coming up on Tuesday. We're going to be taking on a team that's on a fantastic run and it's going to be very difficult," said Ancelotti.
Return to Bernabeu
Playing at home for the first time this season as renovation work continues on their iconic ground, Madrid had a lucky early escape when Nabil Fekir appeared to be fouled in the area.
The incident went unpunished and was not reviewed, but Fekir injured him hamstring and had to be replaced.
Madrid promptly inflicted a sucker punch as Vinicius raced onto David Alaba's ball over the top, lobbing Betis goalkeeper Rui Silva for the opening goal.
Betis, who began the season with wins over Elche, Mallorca and Osasuna, hit back when Borja Iglesias laid off a throw-in for Canales to drive through the legs of Thibaut Courtois.
Vinicius should have immediately restored Madrid's lead, blazing over a cross from Rodrygo. Silva then made a miraculous stop to claw out Aurelien Tchouameni's header.
Karim Benzema skewed over and then failed to convert from point-blank range moments after half-time, the Frenchman ultimately denied by a crucial touch from Silva.
The Betis keeper again got his fingers to a powerful effort from Luka Modric, but Rodrygo had the final say when he slid home Fede Valverde's pass on 65 minutes.
Atletico Madrid drew 1-1 at Real Sociedad as Nigeria striker Umar Sadiq scored on his debut for the hosts.
Alvaro Morata swept in at a corner in the fifth minute, but Sadiq headed home a second-half equaliser following his arrival from Almeria.
The 25-year-old joined Sociedad on transfer deadline day as a replacement for Sweden forward Alexander Isak, signed by Newcastle last week for a reported £58 million ($68 million).
More concerningly for Atletico, goalkeeper Jan Oblak was forced off with a leg injury late on.
"Hopefully it will be minor and the doctors will tell us in the next few days," said Atletico coach Diego Simeone, whose side are at home to Porto next week in the Champions League.
Mallorca and Girona drew 1-1 in the early game.