The royal court said in a statement that Abdullah, believed to be around 90, died at 1:00 am local time expressing its "great sadness and mourning".
Salman, 79, had been defence minister and previously governor of the capital Riyadh.
Another of the late monarch's half-brothers, Moqren, was named the new crown prince.
Abdullah will be buried later Friday following afternoon prayers, and citizens would be invited to pledge allegiance to the new monarch and the crown prince at the royal palace, the statement said.
The royal court did not disclose the cause of Abdullah's death, but he was hospitalised in December suffering from pneumonia and had been breathing with the aid of a tube.