The German duo needed nearly three-and-a-half hours to seal a 7-6 (7/5), 6-7 (9/11), 7-6 (9/7), 6-3 win over Spain's Fernando Verdasco and David Marrero in the World Group clash in Frankfurt.
Kohlschreiber and Florian Mayer had given the hosts a 2-0 overnight lead.
The Germans will now face France away in the last eight on after the French took a 3-0 lead over Australia in their tie in Mouilleron-le-Captif.
"We managed to beat a top double pairing, you can't get better than that really," said 35-year-old Haas.
"There were so many emotions, that was unbelievable."
In a tight doubles match, the first three sets all went to tie-breakers with the Spanish pair battling to keep their team in the tie.
The guests were without world number one Rafael Nadal and David Ferrer, ranked fifth in the world.
Kohlschreiber stepped in for Daniel Brands in a tactical switch and the move paid off allowing Germany to reach the quarter-finals for the first time since 2011.
In Friday's singles, Kohlschreiber got the hosts off to a good start in Frankfurt with a comfortable 6-2, 6-4, 6-2 win over Roberto Bautista Agut.
Mayer then dug deep to win a five-set thriller against Feliciano Lopez, needing nearly four hours, to win 7-6(7/6), 7-6(7/4), 1-6, 5-7, 6-3 to put Germany 2-0 up.
Left-hander Lopez fired down 29 aces, but Mayer held his nerve and pushed through the pain barrier.
He later revealed he had needed medication for a painful shoulder just during the match.
Germany will now be hoping to avoid a repeat of their 2011 Davis Cup quarter-final when they lost 4-1 to France in Stuttgart.