The latest Nielsen data shows year-on-year sales at Asda up 8.7% for the 12-week period to 1 November, compared with an average 5.1% growth for the grocery market overall.
"Asda now has the strongest momentum, with total till growth in excess of 8% since the end of August," said Mike Watkins, senior manager retailer services at Nielsen.
"Impulse is suffering at -3.2% year-on-year with shoppers doing fewer top-up and small shops."
Asda this week published third-quarter results showing like-for-like sales up 6.9%.
Morrisons has also grown market share over the past year, increasing it from 9.8% to 10.3%. Its sales increased 9.1% in the 12 weeks to 1 November.
The figures will come as a blow to the wholesalers and independents who hoped more shoppers would abandon their cars in favour of walking to local shops as the recession hit home.
At Tesco, market share has grown 0.1 percentage points to 28.1% year-on-year, with sales up 4% in the past 12 weeks.