Randall announced he was leaving the BBC in September and has since joined the Daily Telegraph to write a twice-weekly business column as editor-at-large.
Peston, who was only named an associate editor two months ago, is the former City editor of the Sunday Telegraph and will only be the second person to hold the job of BBC business editor.
The role was created for Randall, a former City editor of The Times, as well as editor of Sunday Business, now The Business, four years ago.
Randall's appointment saw the BBC shake up its business news output across television, radio and the internet, creating the BBC Business and BBC Money brands as well as a dedicated business reporter on 'Newsnight'.
It also saw the introduction of regular evening business programme on BBC News 24 and a revamping of flagship BBC business show 'The Money Programme' to tackle one subject in depth each week.
The importance of the BBC business editor's job, which is seen as the most high-profile in financial journalism, will put him on a par with BBC political editor Nick Robinson.
Peston will appear across the BBC, including the BBC One's flagship news programme, 'The 10 O'Clock News' and on BBC Radio Four's 'Today' programme. He will also be heard on Radio Five Live news and will be responsible for helping the BBC develop its business coverage.
Prior to joining the Sunday Telegraph in 2002, Peston spent time at the Financial Times and more recently he became editorial director of online financial analysis service Quest in 2000.
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