Barca fell behind in the 31st minute after a moment of madness from Brazilian keeper Neto, who clumsily allowed a back pass from Gerard Pique to trickle through his legs towards goal, and Luis Rioja nipped in to nudge the ball into the empty net.
Alaves midfielder Jota was sent off in the 62nd minute for a second booking and Antoine Griezmann levelled moments later but while the Catalans kept on pouring forward they failed to take their flurry of late chances.
Barca, who were beaten 3-1 by Real Madrid last week, have taken only two points from their last four league games and Ronald Koeman's side sit 12th in the standings with eight points from six matches -- the club's lowest point total after six matches since the 2002-03 season.
"We are very angry because we really wanted the three points and we have to improve a lot," said Griezmann.
"We had a lot of chances but only scored once. 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.
"We are not doing well, we need to improve and with hard work we will but we should stay calm because it's a long season."
The Catalans had produced their best display under Koeman in Wednesday's 2-0 win at Juventus and made a strong start here, wasting an early chance to take the lead when Ansu Fati entered the area but fired wide of the far post.
Griezmann also missed the target in the first half while Lionel Messi saw a free kick cleared off the line.
Alaves responded and Neto showed a sign of what was to come when he spilled a tame shot at the feet of forward Edgar, who missed the target.
The goalkeeper's catastrophic error saw Barca fall behind for the fourth league game in a row and they took their time to regain their composure.
They equalised with a cold-blooded finish from Griezmann and continued to lay siege to the Alaves goal but were repeatedly thwarted by former Real Madrid keeper Fernando Pacheco, who denied Messi, Ansu and substitute Sergino Dest.
Griezmann and Messi, meanwhile, had goals ruled out for clear offsides.