Fogarty, who has presented 5 Live Breakfast with Nicky Campbell since January 2004, will take over the weekday 12pm-2pm slot from Logan in early May.
The show airs a mix of the latest in news, sport and politics from around the world, including the weekly Prime Minister's Questions from Westminster.
Logan took on the programme earlier in the year, but will step down next spring due to family and other work commitments.
Discussing her move, Fogarty said: "I'm in the lucky position of leaving a job I love to start one I'm really excited about. Nicky says he forgives me but I don't believe him!
"Hopefully he and I can still do programmes together in the future one way or another. I learnt a huge amount just by sitting next to him, and I'll miss him."
Radio 5 Live controller Adrian Van Klaveren added: "Shelagh's done an outstanding job on breakfast for the last seven years, playing a pivotal role in helping the programme to win many awards including Sony Golds.
"Her on-air relationship with Nicky has been magical and it's not a partnership to be broken up lightly - but from next May she will be able to bring her own individual magic to lunchtimes and create a must-listen programme, presented at first from London but then of course from Salford by the end of next year."
The BBC said that a new co-presenter to join Campbell on 5 Live Breakfast "will be announced in the New Year".