Germany threw away a 4-0 lead in the final half an hour of the game to claim only a point which still keeps them three points clear at the top of qualifying Group C.
However, opponents Sweden have the chance to join them with a win in their game in hand.
"We have got to face up to this painful situation and not just go back to normality," said Bierhoff.
"You always learn from negative experiences. I hope that we do better in the future and that we become more complete through further experiences."
Bierhoff refused to pin the blame on any one individual, but said it was a collective failure not to hold on to all three points having established such a lead.
"We have experienced players in our side who play in Europe for their clubs," he told Sport1.de.
"We just have to learn to keep a level head in difficult moments - we failed to do that today [Tuesday].
"We made the same mistake in the final 30 minutes that we did against Austria. We played the ball back to Manuel (Neuer) rather than keeping possession or, like at the very end, taking it to the corner flag.
"But I realised during my career and in my life that these ugly experiences help you in the long term.
"I hope this was an important lesson and that it is clear to the players they will never make these mistakes again."