The figures, supplied to The Times, show in the 12 weeks to November 2 that £22m of spend was switched directly from Tesco to Asda, just over £10m to Morrisons, and almost £10m to Aldi.
However, there was minimal switching to Lidl, which together with fellow discounter Aldi has been seen as one of the main threats to the UK's big four supermarkets.
The period measured started six weeks before Tesco introduced its 'Britain's biggest discounter' campaign highlighting its launch of around 400 discount brands.
The figures have emerged on the day that Tesco reported its latest quarterly figures, showing UK revenue growth (excluding new stores and petrol) has slowed to 2% year-on-year. Worldwide revenues climbed 11.7%.
Tesco said the impact of the new discount ranges had taken two to three percentage points off its UK sales margin but had boosted volumes by attracting extra customers.
The TNS figures also indicate that sales growth at Tesco, at 5.4%, is lagging the other big three supermarkets -- Morrisons was up by 9.4%, Asda by 9% and Sainsbury's by 6%.