philipp lahm, sami khedira, miroslav klose and marco reus struck to send the 2004 winners home, though greece were briefly level in gdansk.
greece's hopes of repeating the miracle of 2004 vanished with a 4-2 defeat against germany, who marched on to the uefa euro 2012 semi-finals on the back of a fourth straight victory at these finals.
germany took a deserved 39th-minute lead through philipp lahm and although giorgos samaras's equaliser ten minutes after the restart hinted at something unexpected, joachim löw's men reasserted their authority as sami khedira, miroslav klose and marco reus all found the net prior to dimitris salpingidis's late spot-kick.
it was a world-record 15th consecutive competitive win for a german side bidding to end a 16-year wait for a major honour, and should leave their confidence levels sky-high entering their warsaw semi-final against england or italy next thursday.
for greece, by contrast, this is the end of the road. they had entered the game without the suspended giorgos karagounis and josé holebas, and with worries over the fitness of left-back giorgos tzavellas – hardly the most promising backdrop to their biggest game since the final of uefa euro 2004, a victory famously overseen by a german coach, otto rehhagel.
that summer they kept upsetting the odds but in the first half there seemed zero likelihood of any surprise against opponents they have still never beaten. germany's vibrant display certainly justified the team selection of coach joachim löw, who had sprung a surprise by introducing reus and andré schürrle in the wide positions previously occupied by thomas müller and lukas podolski, as well as klose for mario gomez at centre-forward.
löw's plan was to provide some unpredictability along with speed and mobility and it worked, with reus particularly impressive. the first scare for greece came when michalis sifakis failed to hold khedira's shot, and klose turned in the rebound only to be flagged offside. if sifakis collected a cut to the head in the process, very soon the whole greece team had a headache as germany, watched by chancellor angela merkel, dominated.
the 23-year-old reus had already sent one shot wide when, midway through the half, a slick interchange between him and klose ended with mesut özil testing sifakis. klose and reus both threatened again before lahm struck, with his fifth goal in 90 appearances. collecting özil's pass 25 metres from goal he cut inside and sent in a swerving shot that brushed sifakis's fingertips on its way into the far corner.
greece have been slow starters in this tournament but they came to life at the start of the second half, equalising when salpingidis put a low ball across goal and samaras got there ahead jérôme boateng to squeeze it under manuel neuer.
greece had done something similar to poland in the opening match, but here they were unable to hold on to parity. within six minutes, boateng crossed from the right and khedira crashed a spectacular volley high past sifakis. germany would not let their lead slip a second time and klose duly made sure by outjumping kyriakos papadopoulos and the flailing sifakis to head the third.
when sifakis blocked a klose effort, reus smashed in the rebound for germany's fourth. greece were out but salpingidis's late penalty after a boateng handball at least gave their fans one final thing to shout about.