Dick Advocaat's side made a slow start but soon found their stride to take a 2-0 interval lead through midfielders Dzagoev and Roman Shirokov.
Vaclav Pilar gave the Czechs hope by pulling one back shortly after the break but Dzagoev and substitute Roman Pavlyuchenko, on for the profligate Aleksandr Kerzhakov, sealed the win.
Despite Russia being favourites to top the group, it was the Czechs who settled the better of the two sides.
It took Russia almost a quarter of an hour to muster a meaningful attack, and when they did they came close to taking the lead.
The imperious Andrey Arshavin brilliantly played in Yuri Zhirkov near the left-hand byline but striker Kerzhakov could only steer his effort wide.
But 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 that was headed against the post by Kerzhakov, Dzagoev followed up to slam home the loose ball.
The CSKA Moscow midfielder should have doubled the lead just a few minutes later after another incisive Russian break out, but he flashed wastefully wide of Petr Cech's goal.
Rezek saw a glancing header saved by Vyacheslav Malafeev at the other and but an error from Jaroslav Plasil allowed Russia to make it 2-0 in the 24th minute.
The midfielder gave away possession and Arshavin seized on it, the Arsenal player's pass was meant for Kerzhakov but the striker's miscontrol saw the ball reach Shirokov, who lofted over the advancing Cech.
Plasil was fortunate to get away with a clumsy push on Arshavin in the penalty area, as English World Cup final referee Howard Webb waved away the appeals.
Kerzhakov was once again guilty of being profligate in front of goal 12 minutes before the break, blazing over the bar from Zyryanov's pull back as Russia again toyed with the Czech defence.
Having been second best for much of the opening stanza the Czechs were again on the back foot after the restart with Kerzhakov again going close.
But Russia's momentum was halted in the 52nd minute as they were pegged back by by Pilar.
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 Russia as the match became increasingly open.
Arshavin's artistry meant Russia remained a threat at the other end, but once again Kerzhakov failed to hit the target from a lovely through ball.
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, before the Arsenal midfielder's awkward angled shot was gathered at the second attempt by Malafeev.
But Dzagoev effectively killed the game off when Pavlyuchenko seized on a loose ball 12 minutes from time to set up the midfielder to plant a firm shot beyond Cech.
Pavlyuchenko then put a gloss on the win as he tricked his way into the box before firing into the roof of the net.