Petr Jiracek and Vaclav Pilar scored their goals inside the opening six minutes as Michal Bilek's side bounced back from their 4-1 defeat to Russia in their opening game.
Surprise Euro 2004 winners Greece were handed a lifeline when Theofanis Gekas capitalised on a Petr Cech blunder eight minutes after the break but they were unable to force an equaliser and now face a struggle to reach the quarter-finals with one point from two games.
The Greeks were without two key defenders, Avraam Papadopoulos, out of the tournament with a knee injury, and Sokratis Papastathopoulos, who was suspended, and the Czechs raced into the fastest 2-0 lead in European Championship history.
Wolfsburg wideman Jiracek nipped in ahead of left-back Jose Holebas to slot home the fourth fastest goal in European Championship history after just 134 seconds following a slide rule pass from Tomas Hübschman.
Keeper Kostas Chalkias got a hand to the effort but couldn't keep it out and he was also culpable for the second as was floundering defender Holebas.
Czech right-back Theodor Gebre Selassie got to the by-line with ease and his cut back went under the hand of the one-time Portsmouth custodian and was smuggled into the back of the net by Pilar.
Viktoria Plzen youngster Pilar was almost on his knees as he got between Vassilis Torosidis and Kostas Katsouranis to bundle the ball home.
Chalkias then went off injured and his replacement Michail Sifakis pushed a Tomas Rosicky effort past the post with the Arsenal man plus dangerous wingers Pilar and Jiracek off-target with further first-half attempts.
It was scarcely believable that Greece had lost only once in 22 matches under coach Fernando Santos on their first-half showing but they could have pulled one back when Georgios Fotakis nodded home only for the offside flag to be raised by the narrowest of margins.
Greece clawed their way back against Poland for a 1-1 draw in the tournament opener after conceding an early goal and they were back in the game on 53 minutes thanks to a howler from Chelsea custodian Cech.
A harmless ball from Georgios Samaras into the area was fumbled by Cech, who raced off his line but then turned his head away as he feared a collision with Tomas Sivok, and Gekas passed the ball into the empty net.
The Czechs' attacking threat relented and Cech must have been fearing a repeat of Euro 2008 when his mistakes saw his country eliminated after a late capitulation against Turkey but they were rarely put under pressure by a team that netted just 14 times in 10 qualifying matches, the lowest tally among the finalists.
Sivok blocked Kyriakos Papadopoulos' shot and Giorgios Karagounis teed up Giannis Maniatis whose 30-yard strike was well wide.
But they failed to carve out a real chance in open play and did not look like getting behind the Czech full-backs as they relied on high balls from deep which were comfortably dealt with by Sivok.
The Czechs went to Cech at full-time as they recorded their first win and will now be hoping for a repeat of 1996 when they reached the final despite losing their opening match.