The world No. 1 was rarely troubled by 50th-ranked Haase on the temporary clay court in Amsterdam as he comfortably sealed victory and his team's return to the World Group next year.
"I felt like I was always somewhat in control," Federer said, in an understatement.
With the outcome of the playoff already sealed, Thiemo de Bakker beat Stanislas Wawrinka 6-2, 7-6 (4) to make the final score 3-2 for Switzerland.
Federer and Wawrinka won their opening singles on Friday, and lost in doubles on Saturday to Haase and Jean-Julien Rojer.