Machin's side were comfortably beaten at Butarque on Sunday to leave them five points from safety and now have just two points from their last eight games.
The entire coaching staff of Jordi Guerrero, Jordi Balcells and Carles Martínez have all also departed the Catalan club.
Abelardo took over at Alaves in December 2017 and had worked wonders at the Vitoria-based club, initially leading them to safety before an 11th-place finish last season.
Alaves were holding a top five spot until November and were challenging for a European place until early spring, and only slipped out of the top half of the table on the final day.
Abelardo had been linked with replacing the outgoing Quique Setien at Real Betis while other reports suggested he could replace Luis Enrique - but Rubi and Robert Moreno respectively took those posts at the time.