The group increased its market share to 31.4 per cent for the three months to June 18, compared with 31.1 per cent last month and 30.1 per cent a year ago.
The data also revealed that sales are up 2 per cent at the Morrisons chain - the first increase since its acquisition of Safeway. However, its market share remained unchanged at 11.3 per cent, still down on last year's 11.7 per cent.
TNS director Chris Longbottom said: 'The 2 per cent increase in value clearly demonstrates that the business is turning the corner and seems poised for market share growth.'
Morrisons reported its first annual losses in the company's history in March this year.
Asda kept the number two spot, increasing its market share to 16.5 per cent from 16.4 per cent last month, compared with 16.3 per cent a year ago.
Sainsbury's, which had been gaining ground on Asda, maintained a market share of 16 per cent.
Longbottom said: 'A feature of this period's trends has been the buoyant performance of all the industry's leaders.'