Alaves appointed Asier Garitano as coach on Tuesday after Abelardo left at the end of his contract.
Garitano, who took Leganes into La Liga in 2016, joined Real Sociedad last summer but was sacked in December after eight defeats in 17 league games.
“Deportivo Alaves want to welcome the coach from Vergara and wish him the best of luck in his new era as a blue-and-white,” Alaves said in a statement.
Alaves finished 11th this season but were in contention to qualify for European competition for much of the campaign.
“Alaves did their utmost to keep hold of me,” Abelardo told reporters on Monday.
He added: “I know myself better than anyone and knew I couldn’t reach the same heights as I have this season, which is why I chose to leave now.”