Fogarty, who has presented the breakfast show with Campbell since January 2004, will take over the weekday 12pm-2pm slot from early May, replacing Gabby Logan.
Burden, the co-presenter of 5 Live's Weekend Breakfast since 2003, will succeed Fogarty as Campbell's co-host on the weekday 6am-10am programme, starting on May 3.
Commenting on her new role, Burden said: "It's a brilliant opportunity to join what is already a hugely successful team. Shelagh is the mother of all breakfast broadcasters and I can only hope to match the impression she's made on the early mornings of millions of people.
"My family is already gearing up for the move to Salford and I am really excited about working alongside Nicky and the rest of 5 Live in our new home in Salford Quays."
Welcoming his new co-host, Campbell added: "Rachel's a great broadcaster, a great journalist and a great friend. It's going to be great."
Burden's appointment comes after 5 Live registered its best ever audience figures in the final three months of 2010, attracting more than 7m listeners every week, according to the latest RAJAR data.
"Rachel is already hugely popular with our listeners and is a fantastic journalist who will be a wonderful addition to the 5 Live Breakfast team," said Adrian Van Klaveren, controller of Radio 5 Live.
"She's someone who loves radio and instinctively understands 5 Live's audience. She and Nicky already know each other well and I'm sure they'll be a winning combination every morning."
Yesterday, the BBC secured radio rights to the 2012 London Paralympic Games, meaning full coverage will be available on Radio 5 Live and digital spinoff 5 Live Sports Extra.