The Toffees were staring down the barrel at a fourth successive top-flight defeat when Alexander Isak put the Magpies in front on the 15-minute mark with a composed finish beyond Jordan Pickford.
However, Everton were handed a huge lifeline late in the second half when referee Tony Harrington pointed to the penalty spot following a review at the pitchside minior after Magpies defender Paul Dummett needlessly wrestled Ashley Young to the ground.
Dominic Calvert-Lewin was tasked with taking the resulting spot kick in the 88th minute, and the sheer pace and power of his strike from 12 yards prevented Martin Dubravka's outstretched right hand from stopping the ball arrowing into the top corner.
The Toffees striker celebrated his 22-game goal drought across all competitions coming to an end, and Sean Dyche's men eventually held on for what could prove to be a valuable point in their quest to avoid relegation.
Everton's winless run goes on
However, Everton have now gone 13 games without a victory in the Premier League for the first time in their history, a run that includes seven defeats and six draws.
The last time the Toffees endured a winless streak of this length was between October and December 1957, while they have only ever experienced one longer winless run in their top-flight history - 14 games from March to September 1937.
Everton last celebrated success in the Premier League when they beat Burnley 2-0 at Turf Moor on December 16, before a 2-1 defeat at Tottenham Hotspur a week later began their 13-game winless streak.
Two defeats to Manchester City as well as losses against Wolverhampton Wanderers, West Ham United, Manchester United and Bournemouth have taken place during this run, while the Toffees have picked up solitary points in draws versus Aston Villa, Fulham, Spurs, Brighton & Hove Albion, and now Newcastle.
Everton have only scored eight goals across their 13-game winless run, drawing a blank in front of goal on four occasions, while they have conceded a total of 21 goals, keeping just two clean sheets.
Dyche remains upbeat despite winless run
Everton remain 16th in the Premier League table, four points above the relegation zone with eight games remaining, and Dyche believes his side have eared a "good point" on the road against Newcastle.
"I thought it was a pretty decent performance, but at half time I mentioned that edge and how we turn that into a winning performance and I thought second half was a very valid attempt to do that," Dyche told reporters after the match. "We didn't get the win but we get a good point on the road in the Premier League, it's always important.
"The penalty of course, we've only had one this season, should have had another one Saturday but we didn't get that. I don't know why they have to look at that a hundred times, it's absolutely bizarre because I've seen it back and it gets given straight away.
"Anyway, they did get to the right decision so we're pleased with that, and Dom put it away so we're pleased for him as well."
Commenting on Everton's 13-game winless run, Dyche said: "This job wasn't a walk in the park when I got it and it is still not, there is a long way to go but there are a lot of steps we are making in the right direction.
"I know we are capable of getting wins and results, we haven't scored as many as we would like but the mentality is right and certainly right tonight."
Everton return to Goodison Park on Saturday afternoon when they play host to fellow relegation-threatened side Burnley.