The Toffees were staring down the barrel at another defeat after Alexander Isak put the Magpies in front early in the first half with a composed finish.
Newcastle were in control for the majority of the contest, but they could not put the game out of sight and that proved costly, as Calvert-Lewin converted an 88th-minute penalty - his first league goal since October 29 - to salvage a valuable point for Sean Dyche's side.
While Everton can take positives from sharing the spoils with Newcastle, they have now broken an unwanted club record in the Premier League after extending their winless run to 13 games.
The Toffees have only ever had two longer winless streaks in the top flight winless runs - a run of 14 matches from March to September 1937 and a run of 13 games between October and December 1957.
Dyche's men remain 16th in the Premier League table, just one point clear of 17th-placed Nottingham Forest - who comfortably beat Fulham 3-0 at the City Ground - while they Toffees are four points clear of Luton Town in the relegation zone.
As for Eddie Howe's Newcastle, they missed the chance to climb into the top seven and remain eighth in the table, only one point behind seventh-placed West Ham United.
After scoring twice as a substitute in Newcastle's thrilling 4-3 comeback win over West Ham United last weekend, Harvey Barnes was rewarded with a start by Howe, who made a total of four changes to his first XI - Emil Krafth, Lewis Hall and Elliot Anderson also earned recalls.
Newcastle came out of the blocks and Barnes was denied a second-minute opener by former Sunderland man Jordan Pickford, who came off his line and made himself big to make a strong save from close range.
Abdoulaye Doucoure then blazed Everton's first shot on goal over the crossbar, shortly before James Tarkowski headed over a good opportunity from just four yards out after towering above the Newcastle backline during a set-piece.
Just as Everton were beginning to settle into the contest, Newcastle opened the scoring when Barnes played a smart ball over the top for Isak, who kept his composure as the drove towards goal before slotting a side-footed finish into the bottom corner.
Isak has now scored 19 goals for Newcastle in all competitions this season and has netted in each of his last five Premier League home games - only Alan Shearer (15 in May 1997), Andy Cole (eight in February 1994) and Les Ferdinand (six in November 1995) have boasted a longer such run in the competition.
There was no energy in Everton first-half press and they were at times sloppy in possession; Jacob Murphy capitalised on a loose ball in the middle of the park and drove towards goal before fizzing a powerful shot over the crossbar.
Both Doucoure and Isak had decent efforts at either end saves by Martin Dubravka and Pickford respectively towards the end of the half, but it was the Magpies entered the break ahead.
Newcastle were almost gifted a second goal just four minutes after the interval when Tarkowski flicked an attempted clearance onto his own post following a Magpies corner.
Dan Burn then had the ball in the back of the net, meeting an Isak cross from the right to tap home from close range, but VAR intervened to disallow the goal as Isak was caught offside in the build-up.
Everton struggled to get going in the second half, so Dyche opted to make a triple substitution just after the hour mark, with Beto, Doucoure and Andre Onana all replaced by Dominic Calvert-Lewin, Andre Gomes and James Garner.
The latter made an immediate impact and was whiskers away from restoring parity for the Toffees as his curling right-footed shot from the edge of the area struck the inside of the post.
Just moments later, Vitaliy Mykolenko made a crucial goalline clearance to prevent Isak from scoring a simple tap-in from inside the six-yard box following a well-worked move down the left flank involving Barnes.
Everton looked threatening on the counter-attack in the closing stages, but when both Mykolenko and Calvert-Lewin burst forward in two separate moves, Fabian Schar and Burn tracked back well to make two vital interceptions.
More to follow.