Dickerson, who anchors CBS News's Sunday morning show Face the Nation and serves as the network's chief Washington correspondent, will relocate to NY and join CBS This Morning co-hosts Norah O'Donnell and Gayle King on the show this Wednesday.
Rose was sacked in late November following allegations of sexual misconduct.
"Gayle and Norah continue to show tremendous leadership on our morning broadcast each day", said David Rhodes, president of CBS News, in a statement.
Dickerson is expected to continue his duties as Face the Nation moderator.
"Today is our sixth anniversary". He will start on January 10 and will be based in NY. "I've worked alongside John for nearly 20 years, and this is a great way to continue our mission of putting the news back in the morning".
The journalist John Dickerson is a Washington creature to his core: a presidential historian, repository of political trivia, and scion of a glamorous Kalorama family whose party guests regularly included Kennedys, Johnsons and Reagans.
Dickerson joined CBS News in April 2009 as an analyst and contributor to all of the network's broadcasts and platforms.
Dickerson, who has anchored the CBS Sunday political show "Face the Nation" since 2015 and moderated two 2016 presidential debates, will start his new job on Wednesday, the network said in a statement. Before joining CBS, he covered the George W. Bush administration for Time before moving on to write on politics for Slate. CBS THIS MORNING is in its closest competitive position with "Today" in at least three decades and in almost two decades. CBS is third behind NBC's "Today" and "Good Morning America" in viewership but has made steady progress in ratings and reputation since it was launched with Rose, O'Donnell and King six years ago.