Dzagoev and Roman Shirokov gave Russia a 2-0 halftime lead before Vaclav Pilar pulled one back in the 52nd minute to give the Czechs brief hope of a comeback.
Dzagoev replied with his second, in the 80th minute, and substitute Roman Pavlyuchenko added a fourth in the 82nd.
Russia, built around a nucleus of players who made a surprise run to the semifinals four years ago, showed it has the potential to go even further in Poland and Ukraine.
If Euro 2012's opening 1-1 draw between Poland and Greece had more drama than quality, the second group match was a feast of fluent, attacking football.
Russia coach Dick Advocaat said before the match he only considered his team an outsider for the title, but he may have to revise his opinion after the emphatic victory, although Russia is unlikely to meet many defences as accommodating as the Czech back four.
After weathering a tough opening 10 minutes, Russia settled into its fluid passing game and tore apart the Czech defence.
Dzagoev was on hand to smash home the loose ball after Konstantin Zyryanov floated a cross to the far post and Aleksandr Kerzhakov's header hit the woodwork and bounced to the CSKA Moscow midfielder in the 15th minute.
Dzagoev should have scored again minutes later but blasted his shot wide after Kerzhakov found him in space on the right.
Shirokov made no such mistake in the 24th minute when he ran onto a diagonal pass from the left by Andrei Arshavin and cleverly lifted the ball over Petr Cech's despairing dive for his seventh intentional goal.
After Russian chances early in the second half went off target, the Czechs replied as Pilar beat the offside trap to run onto a pass from Petr Jiracek, get around Vyacheslav Malafeev and slide the ball home in the 52nd minute.
And after Kerzhakov missed two more good chances, Czech Republic right back Theodor Gebre Selassie nearly scored a spectacular equalizer in the 72nd with a powerful right foot volley into the side netting and Tomas Rosicky came close with a low shot that Malafeev saved well.
But Russia then stepped up a gear as Dzagoev slammed home another shot after a pass from substitute Roman Pavlyuchenko and then Pavlyuchenko, on for Kerzhakov, held off three Czech defenders and drove a rising shot past Cech.
Russia: Vyacheslav Malafeev, Alexander Anyukov, Alexei Berezutski, Sergei Ignashevich, Yuri Zhirkov, Roman Shirokov, Igor Denisov, Konstantin Zyryanov, Andrei Arshavin, Alexander Kerzhakov (Roman Pavlyuchenko, 73), Alan Dzagoev (Aleksandr Kokorin, 84).
Czech Republic: Petr Cech, Theodor Gebre Selassie, Tomas Sivok, Roman Hubnik, Michal Kadlec, Jan Rezek (Tomas Huebschman, 46), Jaroslav Plasil, Tomas Rosicky, Petr Jiracek (Milan Petrzela, 75), Vaclav Pilar, Milan Baros (David Lafata, 85).