Nugent, 50, will appear on the flagship morning programme from Monday to Wednesday, joining the line-up of Sheffield’s Dan Walker, plus Naga Munchetty and Charlie Stayt.
She has been filling in for BBC Breakfast's regular hosts on a freelance basis since November 2011 and also presents its sports bulletins.
Writing on Twitter following the announcement, Minchin said: "Brilliant news, massive congratulations Sal. You are a fabulous journalist and wonderful, supportive friend.
"I am so pleased for you and the #BBCBreakfast team."
Born in Birkenhead, Nugent started her career at BBC Radio Merseyside and also worked at BBC North West Tonight and BBC News.
Last year, she followed footballer Marcus Rashford as he sought to end child poverty in the UK, winning scoop of the year at the 2021 Royal Television Society Journalism Awards for her work.
Her other notable stories include an interview with former rugby player Mike Tindall after his father developed Parkinson's disease, and covering fellow rugby star Rob Burrow's journey following a motor neurone disease diagnosis.
She said: "I'm absolutely thrilled to become the new presenter of BBC Breakfast and bring the nation the latest news, top entertainment guests and stories that matter every week.
"It's an honour to take the reins from Louise and be part of the UK's most-watched breakfast show."
Walker, who is competing in Strictly Come Dancing, also sent congratulations.
He said on Twitter: "I am so happy for @sallynugent. She is a brilliant presenter, massively popular with the team, an excellent journalist and a great friend.
"Hopefully we'll have many happy years on the #BBCBreakfast sofa together. Oh... and she's coming to #Strictly this weekend."
Munchetty shared a message from the official BBC Breakfast Twitter account which said: "We are pleased to announce Sally Nugent will permanently join #BBCBreakfast as a Presenter."
Former Breakfast presenter Steph McGovern tweeted: "YES!!! Totally the right choice! Such a brilliant journalist/presenter and all round fab lass. #proudfriend"
Minchin, 53, announced in June that she was stepping back from the show after nearly 20 years.
She first appeared on the flagship morning show in 2001 and moved with its red sofa to Media City in Salford as a lead presenter in 2012.
She was joined by father Patrick, husband David, celebrity friends and colleagues for her final broadcast in September.
The editor of BBC Breakfast, Richard Frediani, said: "Sally has an impressive track record as an award-winning broadcast journalist and has secured a series of high-profile exclusive interviews in recent years.
"We are delighted to have her take her new permanent place in the studio alongside Dan, Charlie and Naga."