Last week Sainsbury offered 305p a share for Low. This was to counter a 225p offer by Tesco a fortnight earlier that put a pounds 155m price tag on the company and was recommended by the Low board.
Sainsbury said it had noted the increased Tesco offer but would not increase its own, which it considered 'full and fair'.
'Sainsbury's firmly believes it is in the strongest position to unlock the potential of William Low's stores but it will not pursue an acquisition at a price that does not provide an acceptable return to shareholders.'
The board of Low, which has been struggling to compete with larger rivals, said it was delighted with Tesco's offer and would consider a formal recommendation.
The stock market was wrong-footed by Sainsbury's withdrawal. Tesco's revised offer of 360p as dealings began prompted a rise in Low shares to 373p before Sainsbury made its announcement at 9am.
Analysts said that 360p was almost certainly the most either would pay and it was a question of which bidder got there first. Sainsbury had little option but to back down in the face of Tesco's revised offer.
With planning permissions for new superstores likely to become harder to obtain, Low presented a one-off opportunity for an expansion into Scotland by the English supermarket operators.
By the close, Low was 23p higher at 359p, Tesco 5.5p lower at 237.5p and Sainsbury unchanged at 419p.
A Sainsbury spokesman said Sainsbury had been in a stronger position than Tesco to benefit from Low. Its own operating profits per square foot were 45 per cent higher than Tesco's, it had much less overlap of stores in Scotland and had advantages in distribution.
Sir Ian MacLaurin, chairman of Tesco, disputed this, saying that takings at Tesco's larger stores matched their equivalents at Sainsbury.
'If what they say is true then J Sainsbury would have been justified in going to 380p or more.'
Prospective returns on its investment fully justified the acquisition which, excluding rationalisation and integration costs, would have a negligible impact on Tesco's earnings per share in the current financial year and make a positive contribution after that.
Tesco plans to retain all 57 Low stores, although there may eventually be relocations in certain towns with a small number of jobs lost at head office.
Its product range is typically up to 9 per cent cheaper than Low's. Average sales per square foot are pounds 16.40 at Tesco compared with pounds 9.60 at Low.
Given this gap Tesco should be able quickly to lift operating profits at the Low chain to pounds 35m from a current level of about pounds 20m, according to Michael Bourke, analyst at Panmure Gordon.
'It is not a particularly exciting investment and I would have preferred it if Tesco had paid less, but it is rather better than putting money on deposit,' he said.
The profits growth would come from cost savings and sales growth as Tesco introduced petrol stations and other innovations at Low supermarkets, he added.
Sainsbury, which is tipped to make a North American acquisition to build on its Shaw's chain, said it would pursue its current 'slow-but-sure' policy of expansion into Scotland, where it had planning permission for one superstore and four recommendations pending for others.
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