Valencia took the lead through Daniel Parejo after Swansea had failed to clear a corner, leaving the midfielder with a simple finish.
Parejo came close again with a low effort that looked to be heading wide, but Tremmel made sure he got a hand on the ball.
Sofiane Feghouli had a great chance to double the visitors' lead after the break, but with just Tremmel he dragged his shot wide of the goal.
Alejandro Pozuelo's overhit cross nearly beat Alves, but the keeper was alert to tip the ball over.
Valencia immediately broke and Andres Guardado came close when his well-struck effort curled just wide of the goal.
Swansea thought they had equalised with three minutes of the game left to play, as Vazquez beat the keeper but saw his strike cleared off the line by Jeremy Mathieu, as the visitors held on to the win.