He was born and died in Christiania (now Oslo). Sylow was a son of government minister Thomas Edvard von Westen Sylow, and a brother of military officer and sports official Carl Sylow. He attended Christiania Cathedral School (1850) and the University of Christiania (cand.real. 1856) .
Sylow was a high school teacher at Hartvig Nissen School, later before becoming a headmaster in Halden from 1858 to 1898. He was a substitute lecturer at University of Christiania in 1862, covering Galois theory. It was then that he posed the question that led to his theorems regarding Sylow subgroups. Sylow published the Sylow theorems in 1872, and subsequently devoted eight years of his life, with Sophus Lie, to the project of editing the mathematical works of his countryman, Niels Henrik Abel. In 1898, he was appointed professor at the University of Christiania.