Reuters reported earlier on Tuesday that Baker would take over from current CEO Helena Foulkes, according to a source familiar with the matter.
Foulkes, who has been at the helm since 2018, will step down effective March 13, Hudson's Bay said in a statement.
"The company and I are grateful for Helena's leadership and significant accomplishments over the last two years," Baker said in the statement. "We are confident in our go-forward leadership team and our ability to drive HBC forward."
The move comes just days after shareholders approved the Canadian department store operator's move to go private in a C$2 billion ($1.5 billion) deal.
Baker and his partners were in a tussle with the company's top shareholders over the deal.
The retail mogul had previously fallen short of necessary votes to win approval for his take-private quest, Reuters reported last year. He later won the support of a significant shareholder after raising the offer price to C$11 per share.
The privatization deal was completed on Tuesday, the company said, and its common shares are expected to be delisted from the Toronto Stock Exchange at the close of trading on March 4.
"This is a great outcome for HBC and all of its stakeholders... It will take patient capital and a long-term view to fully unleash HBC's potential at the intersection of real estate and retail," said Baker.
Hudson's Bay has been shutting stores and selling assets as it sharpens its focus on luxury department store Saks Fifth Avenue and its namesake stores in Canada.
Last year, the company announced the sale of its Lord + Taylor department store business to fashion rental service company Le Tote for about $100 million. It also sold its remaining stake in its German real estate venture last year for C$1.5 billion.
By Melissa Fares