The victory moves Russia into second place in Group G, two points ahead of Sweden after they lost 4-1 at home to an Austria side who clinched their place at the finals in France.
Buoyed by victory at home Sweden on Saturday, which rekindled their qualifying hopes, Russia made a bright and confident start away to the group's minnows.
Zenit St Petersburg forward Dzyuba opened the scoring in the 21st minute when he was put through by an intelligent back heel from Roman Shirokov and managed to place his shot past the despairing dives of two Liechtenstein players on the goal line.
Dzyuba was sent through again after 40 minutes but was bundled over by Daniel Kaufmann after rounding goalkeeper Peter Jehle.
Kaufmann was sent off for a professional foul and Alexander Kokorin scored the resulting penalty to double Russia's lead, sending the ball into the bottom left-hand corner.
Russia made it 3-0 on the stroke of halftime when Dzyuba grabbed his second goal with another close range finish after being put through by Shirokov.
Dzyuba got his hat-trick and Russia's fourth in the 73rd after he got on the end of Dmitri Kombarov's left-wing cross.
Substitute Fedor Smolov then scored from close range with his first touch in the 77th after Jehle did well to keep out Dzyuba's header before Alan Dzagoev curled a free kick over the wall into the top left-hand corner with five minutes to go -- his first goal for his country in over three years.
Dzyuba wrapped up the victory with his fourth of the night and seventh in the qualifying campaign so far.
(Reporting by Dmitriy Rogovitskiy; Editing by Ken Ferris)
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