Allan, currently deputy chair of Dixons Carphone and chair of Barratt Developments and Worldpay, replaces Richard Broadbent who announced his departure in October after the retailer was found to have overstated its profits by £263m.
He takes up his role at Tesco on March 1. Broadbent will step down from the board on the same day.
Allan, who before his retirement in 2009 held a number of roles including chief executive of Ocean Group and chief financial officer at Deutsche Post, will take home an annual salary of £650,000, fixed for three year.
Patrick Cescau, Tesco's senior independent director, said: "Following a deep and thorough process run by a committee of independent non-executive directors, the board unanimously agreed that John Allan was the right candidate to chair Tesco at this important time.
"On behalf of the board I would like to thank Richard for his work as chairman. He has served the business with unflinching commitment through a period of unprecedented change, and put in place a new senior leadership team for the next stage of Tesco's development."
Allan said he was pleased to be taking on this role at "such a critical moment for the business".