The hosts, who move up to eighth in the table, played 65 minutes with 10 men after Mauro Arambarri was sent off for a second booking.
Markel Bergara scored the only goal, with his 25-yard strike deflecting in off Gabriel Paulista.
Dani Parejo almost levelled when his free-kick struck the base of a post.
Simone Zaza, who is the joint second top scorer in the league, missed a glorious chance to equalise deep into injury time, when he headed over from close range.
Elsewhere on Sunday, Leganes moved up to seventh by scoring three times in the last 18 minutes to beat sixth-placed Villarreal 3-1.
Atletico Madrid, who are one point below Valencia, were the only top-four team to win in La Liga this weekend – coming from behind to beat Real Sociedad 2-1 on Saturday.
Barcelona drew 2-2 with Celta Vigo and Real Madrid were held 0-0 by Athletic Bilbao.