The ninth-placed Toffees have now not lost in six Premier League games, five of those since Allardyce took charge.
Captain Phil Jagielka cleared off the line from Tiemoue Bakayoko and Willian in quick succession to frustrate the visitors, who were without suspended top scorer Alvaro Morata.
Chelsea dominated, but struggled to breach the Everton back line and found goalkeeper Jordan Pickford in good form when they did.
The 23-year-old England World Cup hopeful saved smartly from Pedro’s effort in the first half and tipped a low Eden Hazard drive wide after the break.
Substitute Ashley Williams then headed against his own crossbar as the Toffees at times rode their luck to keep a fourth clean sheet in six games.
Antonio Conte’s Chelsea side stay third following the draw, two points behind Manchester United, having lost just once in their past 11 league games.
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