Kohlschreiber had little difficulty in defeating Roberto Bautista Agut 6-2, 6-4, 6-2 in 1 hour, 43 minutes.
Kohlschreiber, the world No. 27, attacked from the start and won five games in a row to take the opening set in 26 minutes. Bautista Agut, who was making his Davis Cup debut, failed to earn a single break point.
Mayer had it much tougher later against Spanish No. 1 Feliciano Lopez, who fought back from two sets down and saved four match points. The German eventually prevailed 7-6(6), 7-6(4), 1-6, 5-7 6-3 in 3 hours, 51 minutes.
The result left five-time champion Spain on the brink of its second successive first-round Davis Cup exit after losing to Canada last year.
Germany needed one more win in the five-match series to complete its first victory over Spain since 1994.
The winner will play France or Australia in the quarterfinals in April.