Alexander Isak's third goal in four days looked set to give Eddie Howe's side a winning start to April, but they were pegged back in the 88th minute by the struggling Merseysiders.
The Magpies had an opportunity to take the lead in only the second minute when Harvey Barnes - handed a start after his double from the bench against West Ham at the weekend - saw his shot well saved by Jordan Pickford.
And that seemed to give the visitors, in desperate need of a victory to climb away from the Premier League relegation zone, confidence. Abdoulaye Doucouré sent a fizzing effort off target in the sixth minute then two minutes later, from a Dwight McNeil free kick, James Tarkowski headed over.
But on the quarter-hour mark, Sean Dyche's men fell behind. Barnes played a brilliant ball for Isak to run onto but the Swedish striker still had it all to do; however, he did just that as he drifted past a couple of challenges and bent the ball beyond Pickford for his 19th goal of the campaign.
Jacob Murphy's powerful strike flew over in the 28th minute before a flurry of chances just before half time.
First, Doucouré turned well but his shot was too close to Martin Dúbravka, then Lewis Hall's deep cross was nodded on by Sean Longstaff towards Murphy at the far post, but his effort was blocked by the legs of Pickford.
And the England goalkeeper was at it again in first-half stoppage time, tipping Isak's piledriver over the bar after United's leading scorer let fly from the edge of the box.
At the beginning of the second half, Everton skipper Tarkowski was relieved to see his sliced clearance from a Murphy corner rebound back off his own post with Pickford beaten.
Then the visitors had another let-off when stand-in skipper Dan Burn tapped home at the Gallowgate End - but VAR cut short Newcastle's celebrations as Isak was adjudged to be offside by the narrowest of margins.
If that was hard luck for United, they did enjoy some good fortune in the 66th minute when substitute Joe Garner struck the base of Dúbravka's post.
A minute later Isak had a shot headed off the line by Vitaliy Mykolenko before Barnes fired narrowly wide, but Everton continued to pose questions as the game headed into the final 15 minutes.
It needed a superb sliding challenge from Fabian Schär to snuff out the danger as Mykolenko prepared to pull the trigger from 12 yards, before Dúbravka pushed away an effort from Ashley Young.
At the other end, Pickford denied Bruno Guimarães before Howe was forced to turn to his bench after what appeared to be yet another injury, with Hall coming off to be replaced by Paul Dummett, making only his second top-flight appearance of the season.
And the long-serving defender found himself penalised for a tussle with Young within four minutes. Both players appeared to have a hold of one another but VAR sent Tony Harrington over to the monitor and the Cleveland-based referee pointed to the spot.
Substitute Calvert-Lewin stepped up and although Dúbravka went the right way, he couldn't keep out the well-struck penalty as United - who had loud appeals for a penalty of their own in added time - had to settle for a draw.