The 22-year-old striker, who headed the winner in Ancelotti’s opening game against Burnley on Boxing Day, shone once again as the Toffees secured a 2-1 success.
Having fired his side in front after just 13 minutes, Calvert-Lewin fumbled what turned out to be winner over the line midway through the second half shortly after Newcastle had hauled level through Fabian Schar.
It proved a painful defeat for Steve Bruce, who restored the likes of Andy Carroll and Jonjo Shelvey having rested them for the Boxing Day thumping at Old Trafford, only to see his side’s attacking intent go unrewarded.
Meanwhile Ancelotti, who made five changes, will have been delighted with his side’s sense of purpose despite a sluggish start in which the hosts were unlucky not to seize the advantage.
Miguel Almiron, fresh from scoring his overdue first goal for the club last week, almost repeated the feat with under a minute on the clock but was let down by his final touch after being teed up by Joelinton.
Four minutes later Everton were in front, Gylfi Sigurdsson delivering a free-kick into the box at the second attempt, where it took a series of minor deflections before landing fortuitously at the feet of Calvert-Lewin to sweep home.
Schar got a vital block on a goal-bound Theo Walcott effort to prevent Everton extending their lead as the visitors, breaking swiftly from midfield, began to assert their authority on the game.
A change in formation around the half-hour mark roused Newcastle, with Almiron looking much more effective in a wider role, and playing a big part in his side’s revival.
Almiron struck the post during a goalmouth scramble in the 29th minute, the ball rebounding for Carroll to prod home from close range only to be ruled narrowly offside.
More pressure from Newcastle saw Joelinton sky an effort from outside the box, before Almiron hit a rising effort of his own which was acrobatically saved by Pickford.
Combining well with Joelinton, Almiron squeezed clear for another run on goal in the dying moment of the first period, but was once again let down by his final touch as his shot drifted wide.
Moise Kean came close to extending Everton’s lead when he drilled a shot straight at Dubravka at the start of the second half, before Newcastle seized a chance to equalise.
Jetro Willems whipped in a free-kick from the left which Carroll cleverly nodded down for Schar to hook the equaliser past a helpless Pickford.
But Ancelotti was evidently impressed with his side’s response, which came within eight minutes after a counter-attack resulting from Carroll losing the ball on the halfway line.
Linking well with Walcott on the right, Richarlison burst clear and crossed low for the unmarked Calvert-Lewin, who slipped but still managed to usher the ball home from point-blank range to ultimately secure the points.