Alba finished off a great team move started by Xavi Hernandez. The Barcelona midfielder passed to Pedro, who flipped on a back-heeled pass for club teammate Alba to round goalkeeper Sergei Veremko and fire home.
Pedro got his turn to make it 2-0 nine minutes later when he received David Silva's through ball before chipping the ball over Veremko.
The Belarus keeper saved defender Egor Filipenko from scoring an own goal early in the second half.
But Spain continued its overwhelming domination, and Pedro doubled the winning margin with two goals in a three-minute span by twice more lifting the ball over Veremko.
Striker David Villa came on late and was close to adding a fifth goal for Spain with a shot that came off the post.
The last time Spain failed to win a qualifying match was in September 2007 when it drew 1-1 in Iceland.
Spain is seeking to win an unprecedented fourth straight major international tournament after repeating as European champions last summer.
Friday's game was not televised in Spain after the producer, Sport Five, failed to reach a deal with any Spanish television channel. This was the first time a Spain match was not broadcast at home since 1983, when Spain visited Malta.