Dzagoev drilled home a rebound to open the scoring after 15 minutes and Roman Shirokov clipped the ball beyond the onrushing Petr Cech nine minutes later to double the advantage.
Russia could and should have increased their lead before the break but the Czech Republic held on to limit the damage.
The Czechs were thrown a lifeline just six minutes into the second half when Vaclav Pilar cut the deficit with a neat finish.
But it proved to be little more than a consolation as Dick Advocaat’s men scored twice more to secure a welcome three points.
Dzagoev bagged his second of the evening 11 minutes from time and substitute Roman Pavlyuchenko added an extra touch of gloss in the 82nd minute.
Despite Russia being favourites to top the group, it was the Czechs who settled better.
It took Advocaat’s side almost 15 minutes to muster a meaningful attack, but when they did they came close to taking the lead.
Andrey Arshavin brilliantly played in Yuri Zhirkov near the byline, only for striker Aleksandr Kerzhakov to steer his effort wide.
It was a brief reprieve for the Czechs as Dzagoev drove at the heart of their defence to tee Konstantin Zyryanov up for a cross.
The delivery was headed against the post by Kerzhakov and Dzagoev followed up to slam home the loose ball.
CSKA Moscow midfielder Dzagoev should have doubled the lead just a few minutes later this time he flashed wastefully wide.
Jan Rezek saw a glancing header saved by Vyacheslav Malafeev at the other end before an error from Jaroslav Plasil allowed Russia to make it 2-0.
The midfielder gifted possession to Arshavin and his pass eventually found its way to Shirokov, who dinked the ball over Cech.
Plasil was then fortunate to get away with a clumsy push on Arshavin in the penalty area and Kerzhakov once again blazed over when well placed.
Having been second best for much of the first half the Czechs were immediately on the back foot after the restart.
Bordeaux midfielder Plasil produced a superb pass to cut the Russian defence, allowing diminutive winger Pilar to round Malafeev and find the net.
Buoyed by the goal, Michal Bilek’s side suddenly looked far more assured and began to take the game to their opponents.
Arshavin’s artistry meant Russia remained a threat at the other end but once Kerzhakov failed to find the target yet again.
The striker’s nightmare evening soon got worse as he shot hopelessly wide after cutting inside Michal Kadlec, prompting Advocaat to put him out of his misery and summon Pavlyuchenko from the bench.
Theodor Gebre Selassie almost produced a stunning effort as his first-time volley from a Tomas Rosicky cross flashed narrowly wide of the near post.
But Dzagoev effectively killed the game off when Pavlyuchenko seized on a loose ball to hand him the opportunity to plant a firm shot beyond Cech.
Pavlyuchenko then put the cherry on the cake as he tricked his way into the box before firing into the roof of the net.
Russia: Malafeev, Aniukov, Berezutsky, Ignashevich, Zhirkov, Shirokov, Denisov, Zyryanov, Dzagoev (Kokorin 84), Kerzhakov (Pavlyuchenko 73), Arshavin. Subs not used: Akinfeev, Sharonov, Izmailov, Kombarov, Granat, Pogrebnyak, Nababkin, Glushakov, Semshov, Shunin.
Goals: Dzagoev 15, Shirokov 24, Dzagoev 79, Pavlyuchenko 82.
Czech Republic: Cech, Gebre Selassie, Hubnik, Sivok, Kadlec, Plasil, Jiracek (Petrzela 75), Pilar, Rosicky, Rezek (Hubschman 46), Baros (Lafata 85). Subs not used: Lastuvka, Suchy, Necid, Limbersky, Rajtoral, Kolar, Pekhart, Darida, Drobny.
Goals: Pilar 52.
Ref: Howard Webb (England).