Germany raced into a two-set lead but with world number one Rafael Nadal roaring on his team mates, Spain fought back to force a fifth set at a packed Plaza de Toros arena.
Spain will meet France in the semis after the holders reached the last four for a third year in a row with Lucas Pouille fighting back to beat Fabio Fognini 2-6 6-1 7-6 (3) 6-3 in Genoa and clinch a 3-1 victory over Italy.
"I played more aggressively as the match went on".
France has now won six consecutive Davis Cup ties, two shy of their all-time record win streak from 2000-2002. It's a great feeling and very unique to play here in front of my people.
Germany's Tim Puetz and Jan-Lennard Struff celebrate winning their quarter final doubles match against Spain's Feliciano Lopez and Marc Lopez.
Ivan Dodig and Nikola Mektic overcame Kazakhstan's Timur Khabibulin and Aleksandr Nedovyesov 6-7 (2) 6-4 6-4 6-2 to hand Croatia a 2-1 lead.
Kohlschreiber also went down tamely in the second set, before putting up more of a fight at the start of the third, but 16-time Grand Slam victor Nadal was relentless and wrapped up the match.
"I didn't even know if I really touched him on the shoulder. We ended up winning a pretty comfortable breaker and from that I felt like we were pretty much on top and we just had to finish it off".
Nicolas Mahut and Pierre-Hugues Herbert had cruised to a straight-sets victory over Fognini and Simone Bolelli in the doubles on Saturday to put France in control, despite Fognini's bad-tempered four-set win against Jeremy Chardy.
World number one Nadal had not played since he retired from his Australian Open quarterfinal against Marin Cilic in January with a leg injury but made light work of Kohlschreiber to draw five-time winners Spain level at 1-1 in the tie.
But Mikhail Kukushkin, the top-ranked Kazakh player at 92, then levelled the tie by seeing off world 28 Borna Coric, 3-6, 7-6, 6-4, 6-2.
Belgium, who are playing without world No 10 David Goffin, do not have a player ranked in the top 100 in their line-up.
In Nashville, John Isner and Sam Querrey lifted the Americans a 2-0 lead over last year's runners-up Belgium as the United States tries to reach the semis for the first time in six years.