In a first half which saw profligate finishing from the 2008 finalists, Philipp Lahm broke the deadlock with a fine long-range strike after 39 minutes. Greece secured a shock equaliser 10 minutes into the second half with Georgios Samaras tapping in a Dimitris Salpingidis cross.
Germany retook the lead after 61 minutes with a sumptous Sami Khedira volley from close-range, before Miroslav Klose headed a third seven minutes later. Marco Reus made it 4-1 with 15 minutes left before Salpingidis converted a penalty a minute before time as Germany secure a semi-final place for the fourth consecutive tournament.
Die Mannschaft began the breathtaking fashion, and were unlucky not to be ahead after just four minutes. A nervy Michalis Sifakis spilled a Khedira effort back into play, allowing tournament debutant Andre Schurrle to fire low into the net and under the goalkeeper before the effort was ruled out for offside.
Fernando Santos' men failed to offer attacking creativity in much of the early proceedings at the PGE Arena - a testament to their game-plan - with a lone shot from Grigoris Makos saved easily by Manuel Neuer. In contrast, Marco Reus fired an effort wide after a brilliant move from midfield at the other end.
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Germany upped their tempo as Greece initiated a period of stability, and it almost paid dividends. Reus played an intricate one-two with Klose before spreading the ball out wide to Ozil. The Real Madrid midfielder could then only fire at Sifakis before the ball was hurriedly cleared by Kyriakos Papadopoulos.
Seconds later, the lively Reus spurned two glorious chances as Joachim Low's side failed to take advantage of their dominance. Firstly, with Ozil pulling the strings on the flank, the 23-year-old swept the ball across goal only to be agonisingly missed by an outstretched Klose, before he sliced another shot wide as Greece floundered.
With the Ethniki seemingly cornered by the Germans, the 2004 winners were not without their own opportunities. A superb ball from stand-in captain Kostas Katsouranis carved open a suspect defence, as Sotiris Ninis forced Neuer into an awkward save, with the goalkeeper only making sure at the second attempt.
Further efforts were subsequently wasted by Low's men towards the end of the first half. A Schurrle shot curled frustratingly wide, before Khedira's fierce drive was again blocked by the impressive Sifakis as a resilient Greek defence held firm.
But Greece's luck would run out five minutes before half-time. Lahm dribbled his way into position outside the area before unleashing a superb swerving strike which deceived Sifakis and nestled in the bottom corner to give die Mannschaft the lead. Minutes later, Schurrle failed with a chance to make it two when he shot inches wide of goal.
The Ethniki were noticeably improving as the game wore on, but were almost 2-0 down after a horror clearance from Sifakis found Schurrle who, instead of shooting, opted to thread a ball towards Klose, only for the move to break down.
At the other end, Katsouranis picked out Samaras with a fine pass, as Greece broke forward with a two against two counterattack, but half-time substitute Theofanis Gekas failed to get sufficient power in his back-heeled pass.
Nevertheless, the Greeks were not to be denied. A quick-fire counterattacking move saw Salpingidis race down the right before squaring a low cross towards Samaras, who positioned himself in front of Jerome Boateng to bundle the ball past Neuer and send the travelling support into delirium.
The celebrations were short-lived, however. Six minutes, in fact, as Germany emphatically recovered to take the lead just past the hour. A speculative cross from Boateng on the flank was spectacularly met with the foot of Khedira, whose volley rifled into the roof of Sifakis' goal.
The Greek resistance completely gave way after that setback, and Germany took full advantage. Ozil's in-swinging delivery from a corner saw Klose rise highest to head home into an empty net after Sifakis opted to commit himself to a punch, as the Lazio forward claimed his 64th international goal.
On another typically German break, Klose latched onto Ozil's neat pass, but could only fire at Sifakis. However, Reus was at hand to lash a shot into an empty net, securing the Germans' record-breaking 15th successive competitive win. Boateng was deemed to have handballed late on, with Salpingidis converting a penalty as Greece exited the tournament on somewhat of a high.
Low's men progress to the semi-final where they will meet either England or Italy on Thursday at the National Stadium in Warsaw.