The chain, whose Christmas advertising campaign features popular children's book character Mog the Cat, saw sales rise 1.2% in the 12 weeks to December 6 and grew its market share to 16.7% from 16.5% a year earlier, according to Kantar Worldpanel.
But Tesco suffered further trading woes as its sales dropped 3.4% in the quarter, while its market share fell to 28% from 29.1% a year earlier.
Embattled Asda and Morrisons were also hit by falling sales - down 3.4% and 2% respectively.
Walmart-owned Asda has lost its position as the UK's second biggest supermarket in recent months, having been overtaken by a resurgent Sainsbury's.
Asda's market share stood at 16.2% in the 12-week period, down from 16.7% a year ago, while Morrisons also saw its share dip to 11% from 11.2%.
Figures also out for the same period from supermarket analysts Nielsen confirm a similar picture, with sales down 3.1% at Tesco, 4.3% at Asda and 2.1% at Morrisons, while Sainsbury's enjoyed a 0.5% increase.
Its data showed the sector as a whole saw sales rise by 0.1% in the 12 weeks - the lowest growth since June and the ninth consecutive month where sales have increased by less than 1%.
Discounters Aldi and Lidl once again saw double-digit sales growth, up 15.4% and 17.9%, according to Kantar.
They now hold a 10% share of the market between them, with their growing popularity driving prices down across the sector as the Big Four try to keep pace.
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