Moonves will remain CEO of CBS, a role he held since 2006.
CBS noted that Redstone's daughter Shari declined an offer to become nonexecutive chairman, but would remain on the board as vice chair.
Sources told CNBC that Viacom's board would meet Thursday to replace Redstone as that company's chairman as well.
Late last month a California judge ruled Redstone should be subject to a mental competency exam requested by his ex-girlfriend, amid an ongoing dispute about whether the 92-year-old billionaire was fit to remain chairman of the companies he controls.
CBS shares rose 4 percent in after-hours trading. Shares of Viacom rose 10 percent.
— CNBC's David Faber contributed to this report