The BBC said Raworth, 53, will present the temporarily-titled Sunday Morning programme from January 9.
In a statement, the corporation said that Raworth will front the popular show for a “short period” while a recruitment process for a permanent presenter takes place.
Once a permanent appointment is made, the programme will relaunch with a new presenter, title and a new look, the BBC said.
Marr’s last show will air on December 19.
Raworth is currently a senior news presenter on the BBC News at Six and BBC News at Ten programmes. She also presents the BBC’s Election Night coverage alongside various other presenters.
It follows Marr’s recent announcement that he is leaving the BBC for rival Global after more than two decades with the broadcaster.
The presenter made the announcement on his Twitter in November saying he was “keen to get my own voice back”.
He added: “I think British politics and public life are going to go through an even more turbulent decade... I have been doing the Andrew Marr show every Sunday morning for 16 years now and that is probably more than enough time for anybody!”
Marr, a former editor of The Independent, joined the BBC in 2000 and went on to present its flagship Sunday politics show from 2005.
He also took over as political editor of BBC News between 2000 and 2005, while also hosting BBC Radio 4’s long-running Start The Week discussion programme in 2002.
In January 2013 the presenter suffered a stroke that left him in hospital for two months. He returned to presenting the Sunday show that September.
He lives in north London with his wife Jackie Ashley, a political journalist and broadcaster.