According to data from Kantar Worldpanel, supermarket sales increased by 0.3 per cent in the 12 weeks to 14 August - the fastest growth in the grocery market since March this year.
There is no evidence yet to suggest the Brexit vote has caused food price inflation, Kantar said. Grocery prices decreased; a representative basket cost 1.3 per cent less than it did last year.
Retail analyst Nick Bubb said the figures "trumpet a revival of fortunes" for the sector, but that "the industry growth is still not exactly Olympic gold medal performance."
Tesco's sales fell at the slowest rate in six months, down by 0.4 per cent. The supermarket's share price was up by 3.39 per cent in early afternoon trading on the news. Sainsbury's and Morrisons' sales declined by 0.6 per cent and 1.8 per cent respectively.
Kantar said promotional sales fell to their lowest level since September 2010, signalling that supermarkets were moving towards more simple price models.
Fraser McKevitt, head of retail and consumer insight at Kantar Worldpanel, said: "Fewer promotions doesn't mean consumers are paying more for their shopping but does reflect the renewed focus on own-label lines which is visible across the market.
"Shoppers are clearly responding to the better value offered through own label rather than money off, with own brand goods growing at both ends of the price spectrum: premium retailer brands are up by nine per cent and value lines up by two per cent."