Svendsen was born in Bergen in 1862.
While a theological student, Svendsen started working as a teacher for the deaf in Christiania. He eventually travelled to Denmark, Switzerland, Italy, France and Germany to make further studies on the education of deaf. In 1895 he was ordained priest for the deaf in Norway. In 1898 he founded the institution Hjemmet for Døve on Nordstrand in Oslo, an institution that was later named after him: Signo Conrad Svendsen center. He edited the magazine De Døves Blad.
The Norwegian priest Conrad Vogt-Svendsen (1914-1973) is his grandson and the sign language expert Marit Vogt-Svendsen (1949) is his great-grandson.
His memorial can be found outside the Conrad Svendsen center (CSS) on Nordstrand in Oslo.