The hosts, who started the day 15th in the table, took a surprise lead when Brazilian keeper Neto was dispossessed by Luis Rioja.
Alaves were reduced to 10 men when midfielder Jota was sent off for a second yellow card in the 62nd minute.
Antoine Griezmann equalised a minute later but Barca could not find a winner.
Ansu Fati was denied a goal on his 18th birthday by Fernando Pacheco's excellent save, while Griezmann had a goal ruled out.
Pacheco frustrated Barca with further saves to keep out Lionel Messi, Miralem Pjanic and Sergino Dest.
While Barcelona top their Champions League group after wins over Juventus and Ferencvaros, they trail leaders Real Madrid by eight points in La Liga after dropping 10 points in six games.
"We are very angry because we really wanted the three points and we have to improve a lot," said Griezmann.
"We are missing a lot of chances, me more than anyone, but it's still early in the season and I know we're going to get better."
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