The CSKA star hit once in both halves as Dick Advocaat's charges ran out worthy winners in Wroclaw, inflicting a punishing blow on their Group A rivals.
Dzagoev and Roman Shirokov hit before halftime to put the Eastern Europeans in a commanding position, although they could have led by more if not for wasteful finishing.
Vaclav Pilar gave the Czechs a shout of getting something from the game with his strike, but Russia saved the best for last as Dzagoev and Roman Pavlyuchenko both hit in the last 15 minutes to seal an emphatic opening victory.
The game started in a scrappy, hectic fashion, with both sides conceding numerous fouls in the first 10 minutes. It was the Czechs, however, who appeared to be in control in the first phases, controlling the ball albeit with very little action in front of goal. Tomas Rosicky fired from long range, but the Arsenal man put his speculative shot high and wide.
Russia was content to sit back at the beginning, but it had demonstrated its ability in front of goal with last week's destruction of Italy. Apparently the Czech defense had not learned the lesson; a fluid counterattack spearheaded by Dzagoev led to Aleksandr Kerzhakov hitting the post, but Dzagoev was on hand to fire home the rebound and give Dick Advocaat's men the lead with just 15 minutes played.
The goal was the cue for Russia to take a stranglehold on proceedings. Dzagoev was played in one on one with Peter Cech following a neat pass from Kherzakov, but the CSKA Moscow star snatched at his effort and could only find the side-netting.
Andrei Arshavin was the inspiration for the second of the evening. The Arsenal star split open the defense with a brilliant pass across the area, which was met by Shirokov to chip over the prostrate Cech and double the advantage. The Czech Republic continued to try and build from deep with the ball at feet, but any mistake was punished by a Russia team that attacked continually with purpose and numbers.
Kerzhakov would have been extremely disappointed not to have registered on the scoresheet during a one-sided first half. The Zenit man had another golden chance to hit the net having been played in from the left-hand side but, stretching and off balance, blazed over the bar. The Czechs meanwhile barely tested Vyacheslav Malafeev at the other end, and would have been relieved to hear the halftime whistle and go into the dressing room just two behind.
Michal Bilek's team came out after the break looking to reduce the deficit, but it was frustrated by a Russian outfit determined to deny its opponent the slightest glimpse on goal. At the other end it remained the side looking most likely to add to the scoreline; Kerzhakov was given another chance on the right-hand side, but once again fired wide.
Seven minutes into the half, however, the Czech side finally found the breakthrough. A wonderful pass from Jaroslav Plasil opened the Russian defense for the first time, and Pilar was the beneficiary as he controlled the ball instantly, took it round Malafeev and slotted into the empty net for a much-needed lifeline. Wroclaw's Municipal Stadium suddenly burst into life with the strike, as the sizeable Czech contingent in the crowd finally found something to cheer after an opening period that had provided little and made themselves heard.
On the pitch as well, the effect of Pilar's effort began to make themselves felt. The Czechs were playing with more calm and assuredness, while the previously flawless passing game from Russia began to look rather more ragged as passes were falling astray. Advocaat's side, however, remained dangerous and should really have sewn up the game with 20 minutes gone of the second half. Kherzakov broke the offside trap perfectly to go clean through, but his torrid evening in front of goal continued as he pulled well wide to blow a golden opportunity.
It just was not to be the Zenit hitman's day. Another decent effort blazed to the left of Cech's post, followed by an attempt on his weaker left foot moments marked the sixth and seventh shots on goal of the evening for Aleksandr, without once forcing a save from the Chelsea goalkeeper. His profligacy meant that Russian hearts were in mouths for the last 15 minutes; a wonderfully-hit volley from Theodor Gebre Selassie from wide on the right momentarily looked like flying in, before ultimately finishing in the side netting.
Pavlyuchenko added a fourth with a high shot into the roof of the net, having bamboozled an exhausted Czech backline.
The victory means Russia goes straight to the top of Group A, overtaking Greece and host Poland after the pair drew 1-1 in the tournament's opening clash.