Victory meant Nadal set the record for the longest Davis Cup winning streak in combined singles and doubles rubbers with 23 matches in a row.
Holders France were tied 1-1 with Italy in Genoa and 2005 winners Croatia were also deadlocked against visitors Kazakhstan after a day of thrilling action on indoor red clay in Varazdin.
The United States are at home to Belgium in Nashville in the other quarter-final, with John Isner due to take on Joris de Loore in the opening singles rubber at 2100 GMT.
Nadal was swiftly back in the swing of things in Valencia, dismantling Kohlschreiber after Alexander Zverev blew away David Ferrer 6-4 6-2 6-2 to put Germany in front.
The 31-year-old Nadal hardly put a foot wrong on his return, having produced 14 of his 30 winners in the opening set of a lopsided contest.
Kohlschreiber also went down tamely in the second before putting up more of a fight at the start of the third, but 16-time grand slam winner Nadal was relentless and wrapped up the match.
"It’s positive of course, winning in straight sets, it was a solid match. I feel comfortable back on the clay,” Nadal said on the Davis Cup website.
"It’s a great feeling to be back, coming back from injuries is always difficult, but it’s great to be in front of my crowd on a very memorable day. I’m very happy with the match."
Lucas Pouille fired the French ahead with a laboured 6-3 6-2 4-6 3-6 6-1 win over Andreas Seppi before home favourite Fabio Fognini redressed the balance with a 6-7(6) 6-2 6-2 6-3 defeat of Jeremy Chardy.
Pouille looked in cruise control after he swept aside Seppi in the opening two sets but the slugfest went the distance after the Italian fought back in front of a passionate home crowd.
The hosts were in danger of falling 2-0 behind as Chardy snatched the opening set tiebreak against Fognini before the 30-year-old Italian raced through the rest of the match to keep the tie in the balance.
Croatia's Cilic breezed past Dmitry Popko 6-2 6-1 6-2 to give the hosts the lead against Kazakhstan but the visitors hit back after Mikhail Kukushkin overpowered Borna Coric 3-6 7-6(5) 6-4 6-2.
