Leonid Slutsky's team took advantage of Sweden's home defeat against Austria to move into second place with two group games left.
Liechtenstein were reduced to 10 men after Daniel Kaufmann was deemed to have denied Dzyuba a goalscoring opportunity when he bundled the forward over in the penalty area.
Aleksandr Kokorin calmly sent his spot kick into the bottom left corner, beyond the despairing dive of Jehle.
On the stroke of half time, Russia wrapped up victory as Dzyuba latched on to Shirokov's lofted through-ball to score.
Jehle made a series of saves from Alan Dzogoev and Shriokov, but was unable to keep out Dzyuba, who completed his hat-trick with 17 minutes remaining.
Fedor Smolov then got on the scoresheet for Slutsky's side before Dzagoev added a sixth with five minutes left.
There was still time for Dzyuba to score his fourth of the evening as Russia finished with seven and move two points ahead of Sweden in the standings. body check tags ::