Melzer's victory gave Austria an insurmountable 3-1 lead and set up a quarterfinal tie against defending champion Spain in April.
Two-time champion Russia lost its opening tie for a second straight year.
On Friday, Melzer won Austria's first point against Igor Kunitsyn and Bogolomov, who made his Davis Cup debut, lost against Andreas Haider-Maurer. Russia won Saturday's doubles.
Haider-Maurer and Kunitsyn were scheduled to play the meaningless final singles match later Sunday.