Luis Rioja put Alaves in front at Mendizorrotza after an error from Barca goalkeeper Neto, with Antoine Griezmann equalising in the second half, a minute after Jota Peleteiro was sent off.
But with almost half an hour left and a extra man, Barcelona were still unable to find a winner, instead settling for a draw to go with recent defeats by Getafe and Real Madrid, and another draw against Sevilla.
It means Madrid, who had earlier won 4-1 at home to Huesca, now sit eight points clear of their Catalan rivals while Atletico are six ahead after they beat Osasuna 3-1. Atletico will go top if they win a game in hand.
Barca pushed for a winner towards the end with Piqué having a header cleared off the line and Messi firing just wide.
The result means Barcelona are 12th with Alavés in 13th.
October 31, 2020