Market researcher Kantar Worldpanel said grocery sales rose 4.8 percent in the 12 weeks to December 25.
That lagged a 5.9 percent increase in food price inflation, suggesting Britons are cutting back on the number of groceries they buy and/or switching to cheaper alternatives.
Many shoppers are curbing spending as disposable incomes are squeezed by rising prices, muted wages growth and government austerity measures.
Tesco, which reports Christmas sales figures on Thursday, lagged the broader grocery market with sales growth of 3.3 percent over the 12 week period, Kantar said.
The supermarket giant started cutting prices in September in a bid to reverse a drift lower in its market share and predicted its sales would suffer in the short term as a pick up in volume would lag an immediate drop in cash taken at till.
Tesco's 'Big Price Drop' has had an aggressive response from its competitors and put pressure on its (market) share, which has slipped from 30.5 percent a year ago to 30.1 percent, Kantar said.
Asda outperformed the broader market with sales growth of 7.3 percent while J Sainsbury
Elsewhere, upmarket grocer Waitrose continued its run of strong sales growth, but was outpaced by budget chains Lidl, Iceland and Aldi.
(Reporting by Mark Potter; Editing by Elaine Hardcastle)