The Germans, who will have to wait until Sunday to find out their semifinal opponents, have won all five European Championship quarterfinals they have played and they have reached the final on the four previous occasions.
In a one-sided contest, Germany took the lead through captain Philipp Lahm late in the first half and survived a scare 10 minutes after the restart when rank outsiders Greece levelled with a breakaway goal from Giorgos Samaras.
But the three-times European champions, who had never previously scored four goals in a match at the tournament, were in no mood for an upset and three goals in 13 minutes kept them on track for their first title in 16 years.
Greece cut the deficit with a late Dimitris Salpingidis penalty.
"We made it unnecessarily difficult for us," said Lahm, who has scored for his country once every two years since 2004 to take his tally to five. "We had huge chances to go into the lead in the first quarter hour. We got into the lead but gifted it away.
"We were too slow at times and made too many easy mistakes but the important thing is that we are in the semifinals.
For the Greeks, shock 2004 European champions, it was an expected end to their campaign after advancing to the last eight with a surprise 1-0 win over Russia.
"We could have held the ball more and maybe taken a few more chances," said Greece coach Fernando Santos. "But we did not manage to breathe. We did not have the space to breathe. They pressed us all the time."
The match billed as the euro zone battle, pitting debt-ridden Greece against its northern paymaster, was always expected to be dominated by the Germans with the Greeks missing creative midfielder and captain Giorgos Karagounis through suspension and it lived up to its billing from the start.
The Germans, cheered on by their Chancellor Angela Merkel in the stands, started at a high tempo with Andre Schuerrle and Marco Reus, replacing Lukas Podolski and Thomas Mueller, charging down the wings.
But after battering the Greek goal for more than half an hour they had failed to get the early goal they wanted.
Germany coach Joachim Loew, who fielded a more offensive team with Miroslav Klose replacing Mario Gomez, had urged patience and it paid off when Lahm released a trademark swerving shot from 25 metres to beat keeper Michalis Sifakis.
Merkel jumped to her feet to celebrate as her lucky charm tag for Germany worked yet again.
Despite the scoreline the Germans made it hard for themselves with sloppy passing and wayward finishing.
Greece, with their youngest ever squad for a European championship at an average of 26 years, refused to surrender and hit back at the start of the second half as Salpingidis burst down the right and crossed for Giorgos Samaras to level.
But Germany's Sami Khedira volleyed in on the hour to settle any nerves.
Klose headed in another four minutes later with his 64th international goal, leaving him four short of Gerd Mueller's German all-time record of 68.
With the Greeks pummelled into submission, Reus struck in the 74th before Salpingidis converted a spot-kick a minute from time to claim a minor consolation.
