As expected, it was the visitors that immediately took control of the possession, but they had to wait until the 12th minute to carve out their first clear chance of the match. It came for Koke after a clever flick from Costa inside the box, but Albania goalkeeper Etrit Berisha made a smart save.
Spain continued to dominate and had another opportunity in the 18th minute when Vitolo looked to pick out the far corner with a curling effort, but Berisha was on hand to make an even better one-handed stop.
Albania had so little of the ball in the opening 30 minutes that the biggest cheer from the home supporters arrived when Odise Roshi broke down the right and delivered, only for Gerard Pique to rise above Bekim Balaj as the danger was cleared.
Spain continued to dominate possession in the final stages of the first period, but Albania goalkeeper Berisha was in fine form on more than one occasion as he took the pressure off his team with a series of excellent claims, which helped the home side enter the interval on level terms.
Pique headed a Koke corner straight at Berisha as Spain came close once again shortly after the break, before Silva fired wide of the post from distance as Albania retreated into a defensive shape once again.
Spain finally made the breakthrough in the 55th minute when a loose clearance from Berisha fell to Vitolo and the Sevilla winger slipped in David Silva, who squared for Costa to tap home from close range.
Eight minutes later, the visitors had their second of the night as Nolito, who replaced Vitolo on the hour mark, evaded the capture of Napoli defender Elseid Hysaj before finding the bottom corner with a clever finish.
The pace of the match slowed in the latter stages as both managers used their full complement of substitutions, and chances were few and far between late on as Spain comfortably collected all three points.
Next up for the new group leaders Spain, who have now won all six of their international matches with Albania, is a home game against Macedonia next month, while Albania will look to return to winning ways when they host Israel.