With co-hosts Poland drawing 1-1 with Greece in the tournament curtain-raiser earlier Friday, the Russians shoot to the top of Group A and become the team to beat.
CSKA Moscow midfielder Dzagoev, 21, put the Russians ahead on 15 minutes, smashing home a rebound off the post from an Alexander Kerzhakov header.
The second came less than 10 minutes later as midfielder Roman Shirokov chipped deftly over Petr Cech after a through ball from Andrei Arshavin.
The Czechs pulled one back early in the second half through Wolfsburg striker Vaclav Pilar, who calmly rounded Vyacheslav Malafeev and slot home from a tight angle.
Roman Pavlyuchenko came on for Alexander Kerzhakov on 75 minutes, and the former Tottenham man immediately set up Dzagoev for his second before firing in a stunning fourth for his team a minute later.
It was a performance on a parallel with Russia’s victory over the Netherlands at Euro 2008, and hails the arrival on the international scene of CSKA Moscow’s Dzagoev, who has for years been mooted as a star of the future.
Russia takes on Poland in Warsaw for the next game Tuesday while the Czechs will face Greece.
Russia: Vyacheslav Malafeev, Alexander Anyukov, Sergei Ignashevich, Alexei Berezutsky, Yuri Zhirkov, Igor Denisov, Konstantin Zyryanov, Roman Shirokov, Alan Dzagoev (Alexander Kokorin, 85), Andrei Arshavin, Alexander Kerzhakov.
Czech Republic: Peter Cech; Theodor Gebre Selassie, Michal Kadlec, Roman Hubnik, Tomas Sivok; Jan Rezek (Tomas Hubschman, 46), Jaroslav Plasil, Tomas Rosicky, Petr Jiracek (Milan Petrzela, 75), Vaclav Pilar; Milan Baros (David Lafata, 87).