Rosol, who is 63rd in the ATP rankings, won 7-6, 6-7, 7-6, 6-2 in his first ever head-to-head meeting with the 25-year-old Korolev to give his team an unassailable 3-1 lead in the tie.
Hard-serving Lukas fired 31 aces in the match to record his sixth Davis Cup win. The 63rd-ranked Czech landed his 15th ace to prevail on a first-set tiebreaker and his 30th to win the third set.
Defending champions Spain reached the Davis Cup quarter-finals for a sixth successive year on Saturday when they secured an unassailable 3-0 lead over Kazakhstan in their World Group first round tie.