He will run the Tesco board as chief executive Dave Lewis, who joined in September, continues with his plan to fight back from years of market share losses, debt downgrades and the accounting scandal.
Allan is currently chairman of housebuilder Barratt Developments, co-deputy chairman of electricals retailer Dixons Carphone and sits on the board of Royal Mail, but he will step down from the latter two positions when he joins Tesco.
Outgoing chairman Broadbent, in the role since 2011, said in October he would quit, reflecting "the very important principle of accountability". Last year Tesco issued four profit warnings, sending its shares to a 14-year low.
Allan will be paid an annual salary of 650,000 pounds, fixed for three years. He has direct experience of the grocery sector, having worked for Fine Fare in his early career.
"I'm very pleased to be taking on this role at such a critical moment for the business and look forward to working with the new executive team and the board," Allan said in Tesco's statement on Tuesday.
Tesco's new boss Lewis has so far cut prices, improved product availability and increased store staffing levels to help the company to deliver an improved sales performance. He has also detailed plans to slash costs and sell assets to mend the firm's finances.
Tesco's senior independent director Patrick Cescau said that Allan's appointment had been a unanimous decision after "a deep and thorough" selection process.