Page will move into the newly created senior role of managing director, commercial and marketing.
The appointment, originally revealed in May, is a blow to Asda. Page joined the grocer in 1994 and held a number of positions before becoming non-food trading director. Before joining Asda, he was a senior buyer at Sainsbury's.
Woolworths chief executive Trevor Bish-Jones said: 'Tony has a strong track record of driving growth and innovation across a broad range of general merchandise categories.'