Celtic stayed top with 58 points from 20 games, 19 ahead of second-placed Rangers who have played a game more. It was Celtic's 15th league win in a row and ended Rangers' 15-game unbeaten home run.
The home section of a festive New Year's Eve crowd erupted after the 37-year old Miller fired Rangers ahead in the 12th minute, turning in a sharp low cross by James Tavernier after the right back's lovely one-two with Josh Windass.
Sinclair hit the post for Celtic moments before the prolific Dembele netted a 34th-minute equaliser, having been given time and space to chest down a corner and blast a left-footed volley into the roof of the net from 15 metres.
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It was the 20-year old Frenchman's fifth goal in Celtic's third game against Rangers this season, following a hat-trick in September's 5-1 league drubbing and a late winner in October's 1-0 League Cup victory.
The visitors dominated after drawing level and only a string of superb saves by Rangers goalkeeper Wes Foderingham either side of Sinclair's winner denied Celtic a bigger triumph.
Livewire Dembele also hit the crossbar as Celtic kept Foderingham frantically busy but the keeper could do nothing about Sinclair's tap-in from two metres in the 70th minute after Stuart Armstrong's pass across the face of goal.
Miller missed a gilt-edged chance to equalise when he hit the woodwork from close range after a delightful Rangers break in the 84th minutes before the visitors held out comfortably in the closing stages.
(Writing by Zoran Milosavljevic; Editing by Ed Osmond)
