Brydon, chairman of global technology group Smiths Group and the London Metal Exchange, has been appointed for three years from March 26 on a salary of £200,000 a year.
Prior to his role at Smiths Group, Brydon chaired market research group Taylor Nelson Sofres until it was taken over by WPP last year.
In his new role Brydon will be expected to oversee the potential sale of a minority stake in Royal Mail as recommended by Richard Hooper, the former deputy chairman of Ofcom, in a report last month.
Mandelson said: "Royal Mail is now at a crucial stage in its transformation, as it strives to improve efficiency and deliver an even better service to customers.
"I'm delighted to have a man with Donald Brydon's skills and experience to lead the board in this next phase. He is a proven business leader and successful chairman."
Leighton will step down as chairman in March after seven years in the job. During his time as chairman the company has made a turnaround from losses of more than £1m every working day to a position where all four businesses in the group are profitable.
Leighton said: "I warmly welcome this appointment. Donald has a strong track record across a range of business sectors and clearly has the skills and experience to support Adam (Crozier) and the team as they take the business through the next crucial stage of its development."