Germany, it turns out, do not do group stage nail-biters after all. On a clear, mild night in rather desolate low-rise suburban Lviv, a 2-1 defeat of Denmark that was not without alarms ensured it would be Joachim Löw's fluent and hard-running Nationalmannschaft who escaped Group B without the indignity of last-ditch convulsions. Germany will now play Greece, who may feel – in every sense – they owe them one.

With both teams' destiny still in the balance until Lars Bender's late winner knocked the stuffing out of a weary Denmark, this was a fitting conclusion to a group that had been anointed with death-status the moment it was drawn in Kiev. But it had in fact fizzled with unpredictable life, with the fate of all four teams still undecided as the final round of matches kicked off.

Löw is known for a certain restless perfectionism and he was dissatisfied with elements of his team as he looked forward to a quarter-final against a team he described as "very hard – it's like you're biting on a rock".

"It was a very difficult match," Löw said. "In the first half we had three or four chances to make it all clear. We might have killed the situation. In midfield and defence we had too many spaces and Denmark took the tempo out of the game at times. Greece will try to do the same."

Here Germany settled the quicker in front of an agreeably raucous 35,000 crowd, Thomas Müller shooting over the bar after just two minutes, having found space on the right from Lukas Podolski's cross. Chief among this Germany team's shared attributes are a wonderful athleticism and sense of balance in possession and the Podolski-Philipp Lahm axis on the left looked effortlessly penetrative in the early stages.

Denmark, starved of possession, mustered just a sole Nicklas Bendtner shot that was battered well wide of goal before, with a certain inevitability, Germany opened the scoring on 19 minutes. The goal was created by Müller, who skipped inside on the right and crossed for Gomez, who miscontrolled a ball fizzed at his feet. Podolski was in the right place to pick up the deflection and finish from close range. It was his 44th international goal on his 100th appearance.

These are resilient Danes, however, and four minutes later they were level from an expertly worked deep corner, a routine they had already tried once. Bendtner rose to head the ball powerfully back across goal and Michael Krohn-Dehli was alert to nod the ball past Manuel Neuer. Suddenly the match and the group were, if not in the balance, then showing signs of intrigue: as things stood elsewhere Denmark would qualify behind Germany.

Before kick-off there had been some talk of connivance between these two, perhaps even a reprise of the Shame of Gijon, an engineered 1-0 victory for West Germany that kept both themselves and Austria in the 1982 World Cup and eliminated Algeria. Here both teams knew a draw could see them progress to the quarter-finals. Would circumstances conspire to produce a laying-down in Lviv?

Not on the face of it as Germany continued to press, Mesut Ozil's whipped free-kick from the right skimming the eyebrows of Gomez as he tried to force it home from three yards. And three minutes before half-time Gomez almost walked the ball into the net, having picked up a pass from Müller and ambled away from two defenders, only to see Anderson smother. He is the most agreeably quirky of dead-eye goal machines.

Denmark also pressed and exposed a degree of give in Germany's backline as Bendtner won the ball with ease in the air and Germany struggled to clear as ruthlessly as they attack.

With the score in the other match in the group also 1-1 at the start of the second half, there were all possible scenarios to play for. A goal for Denmark might have been enough to put Germany out and they almost had one on 51 minutes, Jakob Poulsen clipping the outside of the post from Bendtner's lay-off. Before long though Germany were showing another side to their game, keeping the ball, at times at walking pace, and drawing the sting of opponents who needed to score. Denmark had one final opening.

In fact, they should have had a penalty after 75 minutes, albeit out of nothing: Bendtner was clearly pulled back by Holger Badstuber stretching for a cross from the left. Nothing was given and Germany exacted a merciless turnaround. Four minutes later Ozil produced a swift diagonal pass to the impressive converted right-back Lars Bender and he finished clinically. Portugal's win over Holland means it is they who progress to play the Czechs on Thursday, a day before Germany's quarter-final.