Dick Advocaat's side could have had more goals on the night, but settled for the four with Dzagoev (15, 79), Roman Shirokov (24) and Roman Pavlyuchenko (82) leading the Czech Republic's pedestrian defence a merry dance.
At 21, Dzagoev is the second youngest player to score twice at the European Championships after England striker Wayne Rooney, who scored his two at the age of 18 in 2004.
Vaclav Pilar scored a fine goal early in the second period to give the Czechs false hope, but they were well beaten by a technically superior side.
Russia will book their place in the quarter-finals with a win over co-hosts Poland on Tuesday night with the Czechs needing a win over Greece to maintain their hopes of finishing second in the section.
They are bottom of the group after the opening day with Poland and Greece sharing a 1-1 draw earlier in the day.
Russia struck the opening goal on 14 minutes when Alexandr Kerzhakov hit the post with a header before Dzagoev followed up to smash the rebound into the unguarded net.
They added a second goal when the lively Andrei Arshavin managed to send a ball spinning through the middle of the Czech Republic defence for Shirokov to dink the ball beautifully over Petr Cech on 24 minutes.
The Czechs somehow dragged themselves back into the reckoning when Pilar bounded on to a ball from Jaroslav Plasil before rounding opposing goalkeeper Vyacheslav Malafeyev to score.
With Kerzhavov missing numerous chances - he is the first player to have seven shots off target in a single game at the Euros since 1980 - before being hauled off to be replaced by Pavlyuchenko on 73 minutes, Russia had to wait until 79 minutes to finally seal the three points.
Pavlyuchenko played provider for Dzagoev to score with a thumping shot from inside the area before Lokomotiv Moscow forward Pavlyuchenko gave the scoreline a truer reflection when he thumped the ball high beyond Cech from just inside the Czech box on 82 minutes.