Finger was born in Bielitz, Austria (now Bielsko, Cieszyn Silesia, Poland) and was educated there. Moving to the United States in 1905, Finger settled first in New Orleans then moved to Houston, Texas three years after. After years in different partnerships, Finger opened his own office from 1923 to 1944.
Finger was the father of Joseph Seifter Finger, a landscape architect and golf-course designer also locally active.
Finger's major work is undoubtedly the 1939 Houston City Hall, designed in a stripped classical style. In response to criticism from Houston mayor R. H. Fonville, who wanted a style with more classical reference, Finger said, "Here in America we are rapidly developing our own type of architecture which is far above that of foreign countries. We are building for the masses, not the classes."
Above the lobby entrance of the City Hall is a stone relief of two men taming a wild horse, symbolizing a community coming together to form a government to tame the world around them. This sculpture, and the twenty-seven other friezes around the building, were done by Beaumont artist Herring Coe and co-designer Raoul Josset.
As the city's foremost Jewish architect in his time, Finger designed many Jewish landmarks, including the 1925 Congregation Beth Israel Temple, now the Heinen Theatre.
Finger's other work includes:
- Texas State Hotel (Now Club Quarters) Houston, 1929
- Montgomery County Courthouse, Conroe, Texas, 1935
- Heights State Bank (later became Rockefeller's, a major Houston music venue)
- Houston Municipal Airport Terminal, now the 1940 Air Terminal Museum
- Harris County Courthouse, Houston, 1953 (with partner Rustay, his last design)
- Barker Brothers Studio, Houston, 1931 now Lawndale Art Center
- James M. and Jessie West Mansion, Houston, 1929