Anand, whose career includes stints at United Biscuits and Sara Lee, before joining Greene King as managing director in 2001, will join the board from 1 January.
His appointment is one of several key hires under new chief executive David Potts as he attempts to turn the struggling business around. Earlier this month the supermarket was kicked out of the FTSE 100 after nearly 15 years, after Morrisons fell to 109th place.
Last month he appointed Post Office chief executive Paula Vennells as a non-executive director, citing her "restructuring" experience as valuable.
Before that he brought in former chief executive of Express Dairies and Arla Foods Neil Davidson. Both firms are suppliers to the supermarket, which is suffering from strained relationships with some of its key partners, particularly in the farming industry.
Chairman Andrew Higginson said: "I am delighted to welcome Rooney to the board. Rooney had the vision at the start of the credit crunch to reinvent Greene King as a branded food based pub retailer. He is a widely admired and successful Plc chief executive.
"His experience in retail, in FMCG and in turning around the fortunes of a major retail business, will all be put to good use at Morrisons. With the recent appointments of Irwin Lee, Belinda Richards, Paula Vennells and Neil Davidson, I am also pleased to say that Rooney's appointment completes the process of renewing and rebuilding the board of Morrisons."
Anand added: "I am excited about joining the team at Morrisons, a company I have long admired. I look forward to working with Andy and the board to help to contribute to the company's return to the forefront of the British supermarket industry."