A 94th-minute goal on Tuesday from Germany striker Cacau sealed the draw with Euro 2012 co-hosts Poland who looked to have secured a first win over Germany at the 17th attempt.
"I am grateful that we can play games like these," admitted Löw, just four days after his team hammered Austria 6-2 to confirm their Euro 2012 place.
"And I am grateful that we got the goal in the final minute and that we don't win every game," he said. "The fact that some things don't work is also my responsibility - if we make so many changes."
"In a friendly, you can't expect everything to work, but it does offer the
coach some insights," he added.
Löw rested vice-captain Bastian Schweinsteiger, play-maker Mesut Özil and goalkeeper Manuel Neuer, opting to play an unusual 4-1-4-1 formation for the first time.
Germany's teenage star Mario Götze and Bayern Munich's Toni Kroos played in the middle with Cologne's Lukas Podolski and Leverkusen's Andre Schürrle on the wings and Miroslav Klose as the lone striker.
But the experiment so nearly back-fired as the visitors struggled to stamp their authority on a plucky Polish side.
Midfielder Jakub Blaszczykowski - known as Kuba for short - converted a penalty in the 91st minute to send Polish fans delirious with delight and make it 2-1 before Germany again came from behind.
In a dramatic finale, German blushes were only spared when second-half replacement Thomas Müller fired in a cross with time almost up to allow Cacau to equalize.
Both Polish goals came from players who ply their trade with German league champions Borussia Dortmund as striker Robert Lewandowski put the hosts ahead on 55 minutes before Kuba's late penalty.
Kroos had put Germany level on 68 minutes when he netted a dubious penalty, but the hosts were reduced to ten men when defender Arkadiusz Glowacki was sent off for a second yellow card on 80 minutes.
Arsenal's 21-year-old goalkeeper Wojciech Szczesny was outstanding throughout for Poland while Kuba, fellow goal-scorer Lewandowski and Cologne's Slawomir Peszko were constant thorns in the German defence.
"It was a very special game," said Polish-born German striker Podolski. "We had some good chances in the first half and if we had scored our goals, things would have looked different. But you have to compliment the Poles, they put up a great fight."