Poor goalkeeping from Kostas Chalkias allowed the Czechs to take command early on in Wroclaw, and despite a blunder from Petr Cech on 53 minutes which allowed Theofanis Gekas to reduce the deficit, they held on to take three points in Group A.
The Czechs' opening 4-1 loss to Russia left them knowing another defeat would end their hopes of reaching the last eight. but they could hardly have wished for a better opening.
Jiracek put them ahead in the third minute as he ran on to Tomas Hubschman's through-ball to fire home, despite Chalkias getting a hand to his left-footed shot.
Chalkias was then at fault for the second goal as he failed to cut out Theodor Gebre Selassie's low cross from the right to allow Pilar to tap in his second goal of the tournament.
The lively Pilar came close to adding a third as he played a neat one-two on the edge of the area before firing just wide of the left-hand post.
Georgios Samaras had Greece's first sight of goal but the Celtic striker made a mess of meeting Vassilis Torosidis' cross.
Chalkias' miserable day continued as he was forced off after just 21 minutes, having pulled up after playing a pass out to his defenders, with Michalis Sifakis coming on to replace him.
Czech captain Tomas Rosicky was booked after his high boot caught his Greece counterpart Giorgios Karagounis on the head but the Czechs' dominance continued as Sifakis was forced into his first save by a low Rosicky drive.
Greece thought they had pulled a goal back five minutes before the break as Georgios Fotakis headed in a Torosidis cross, but he was wrongly ruled to have been offside.
It should have been 3-0 seconds after the restart as the Czechs, whose captain Rosicky was replaced by Daniel Kolar at the break, again surged forward.
Gebre Selassie found more joy down the right but Milan Baros could not get the ball out of his feet when just 10 yards out.
But it was the Greeks who found the net next thanks to a calamitous mistake from Cech.
The Chelsea keeper came racing out for an aimless long ball but, distracted by Tomas Sivok, he lost control and substitute Gekas had the simple task of directing the loose ball into an empty net.
Buoyed by the goal, Greece finally began to exert some control on the game with Karagounis and Samaras becoming increasingly prominent.
But they struggled to create any genuine clear-cut chances, and struggled to test Cech after the Chelsea keeper's blunder, as the Czechs comfortably saw out the closing stages for a vital win.