Everton were sharper from the start and won a penalty when Calvert-Lewin was tripped by Liverpool keeper Alisson
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.