A clever backheel from Arshavin set Yuri Zhirkov free at the byline and his cross found Aleksandr Kerzhakov, but he couldn't direct his side-footed finish on target.
Yet within 60 seconds Russia were in front.
Dzagoev capitaised on two Czechs bumping into each other in midfield to surge forwards and release Konstantin Zyryanov down the right.
He picked out Kerzhakov at the back post and when his header came back off the upright it was Dzagoev who arrived on the scene in the right place and at the right time to drill home.
Dzagoev then wasted a great chance as he took a wild swing with a wood from the green when played in on the right by Kerzhakov.
But on 23 minutes it was two as Arshavin played an incisive ball into the box and although it was an inch too far in front of Kerzhakov, Shirokov sneaked in around the back to dink the ball over Petr Cech.
In Image: Russia's Alan Dzagoyev, second left, and Konstantin Zyryanov celebrate after scoring against Czech Republic