Costa, who opened his Chelsea account in the 2-1 win over Olimpija Ljubljana on Monday, took just three minutes to notch his first assist of the evening at the Gelderdome.
Nemanja Matic sent the Spain international clear and the former Atletico Madrid striker squared the ball for Mohamed Salah to tap into an empty net.
Chelsea doubled their lead 20 minutes later after Costa was felled just outside the Vitesse area. Fellow new-boy Cesc Fabregas curled the subsequent free-kick over the wall and into the top corner of the net.
Jose Mourinho's side went 3-0 up with 15 minutes left to play after another square ball from Costa was met on the volley by Matic.
The home side managed a consolation goal late on when Uros Djurdjevic headed past substitute goalkeeper Mitchell Beeney who replaced Petr Cech earlier in the second half.