Haas and Kohlschreiber defeated Fernando Verdasco and David Marrero 7-6 (7-5), 6-7 (9-11), 7-6 (9-7), 6-3 for a 3-0 German lead in the best-of-five series. Germany will go against France in the quarterfinals April 4-6.
France picked up its third win of the series Saturday against Australia.
Spain, which has won the Davis Cup five times since 2000, was ousted in the first round for the second consecutive year.
Neither side was able to record a break in the first three sets with Germany eking out a 2-1 lead with wins in tight tiebreakers. The lone break of the match was earned by Haas and Kohlschreiber in the fourth set and turned out to be a ticket to the quarterfinals.
Haas and Kohlschreiber won 73 percent of the points on serve and fought back the five break points they faced. Verdasco and Marrero took an equally impressive 71 percent when serving but lost one of the six break opportunities for the Germans.
Kohlschreiber beat Roberto Bautista Agut in straight sets and Florian Mayer added a five-set victory over Feliciano Lopez in Friday's opening singles round to put the Germans on the verge of advancing.
Sunday's schedule calls for Kohlschreiber to play Lopez and Mayer to take on Bautista Agut.