The away side had a chance to take the lead barely two minutes into the match when Cesc Fabregas advanced on goal, but he struck the post. Meanwhile, David Silva also rattled the woodwork three minutes later for the European and world champions.
Spain did have the lead six minutes later, however, when Jordi Alba pounced on a loose ball inside the box to fire his effort past Sergei Veremko. It got worse for the home side after 21 minutes when Pedro latched onto a Silva pass to finish from close range.
The home side's best chance of the first half fell to Vitali Rodionov just before the 30-minute mark, but his effort was pushed wide by Iker Casillas. Spain might have added to their tally late in the first period, but it remained 2-0.
Sergio Ramos was next to come close for the away side after 67 minutes, but his long-range effort missed the target before Cazorla fired wide from inside the box.
Spain eventually had their third and then fourth, however, when Xavi found Barcelona teammate Pedro inside the box and the winger scored his second of the match from close range, before adding his third moments later.
David Villa twice came close to adding to the score in the latter stages, but it remained at four with Spain advancing to the top of Group I.