Three goals in the space of five minutes sparked the first half into life after the 20-minute mark as Adriano and David Villa scored either side of Mario Bermejo's strike.
Jordi Alba wrapped up the win just after the hour mark.
Alba said afterwards: "We knew that we could break the record and we're really happy that we managed to do it.
"I'm a left back, but I like to go forward and it's always nice to score goals. We read the game well after Celta's first counterattacks. We had more control in the second half.
"I feel great, I'm very comfortable on the pitch and I'm sure that I'll improve as the season goes on."