Kamal Ahmed, who began his career in newspapers north of the Border before working for The Guardian and The Observer, is currently business editor at the Sunday Telegraph. He will be seen on TV screens from March.
Mr Peston is becoming the corporation's economics editor, replacing Stephanie Flanders who left broadcasting for a job at JP Morgan.
Mr Peston is best known for breaking the story in 2008 that the Royal Bank of Scotland (RBS), Lloyds TSB and HBOS had been bailed out by the Government.
Mr Ahmed said: "I am absolutely delighted to be joining the BBC. It is a great privilege to follow in the footsteps of Robert Peston and build on the great work the BBC does covering the business world."
The director of BBC News James Harding said Mr Ahmed would "bring intelligence, flair and incisive judgment to the BBC's coverage of business".
Mandarin speaker Carrie Gracie is to be the BBC's first China editor, which Mr Harding said signalled its "determination to bring home to people the sweeping changes in the world's most populous nation as well as the transformative impact that China is having upon the world and all of our lives".