Arnaut Danjuma saved Everton from a fourth successive Premier League defeat as the Dutch forward's equaliser earned a 2-2 draw at fellow strugglers Sheffield United on Saturday.
Sean Dyche's side have endured a dismal start to the season, but a loss at promoted United would have been a new low for the Toffees.
Everton were confronted with that unpalatable proposition when they trailed 2-1 at half-time.
Abdoulaye Doucoure's opener was Everton's first league goal this season, but Cameron Archer equalised and Jordan Pickford's own goal put the Blades ahead.
Danjuma, on a season-long loan from Villarreal, spared Everton's blushes but another grim battle to avoid relegation looks likely for Dyche's men.
Everton sit in 18th while United -- who had also lost their first three games -- are above them only on goal difference.
Everton's lone win this season came against fourth tier strugglers Doncaster in the League Cup in midweek and even that was a barely deserved 2-1 success.
They have won just two of their last 23 away league games and the blue-clad fans packed into one end of Bramall Lane once again cut a disgruntled group at the final whistle.
After two seasons of narrow escapes from relegation, Everton's failure to make any deadline signings on Friday, despite the sales of Alex Iwobi and Neal Maupay, added to the gloomy mood around the club forever mired in turmoil.
After 316 minutes and 46 shots without a goal this season, Everton struck first in the 14th minute.
From Ashley Young's set-piece, Amadou Onana headed goalwards and Wes Foderingham could only parry it out to Doucoure, who converted at the second attempt from close-range after his initial effort was saved.
Doucoure scored Everton's previous league goal against Bournemouth to secure their survival on the final day of last season.
Gustavo Hamer went close to an instant equaliser but the United midfielder's drive drew a fine save from Pickford low to his left.
It was a warning Everton failed to heed as Archer levelled in the 33rd minute.
Archer was making his first Premier League start after his £18 million ($22 million) move from Aston Villa last weekend.
A flowing move climaxed with Oli McBurnie teeing up Archer with a deft pass and the 21-year-old striker arrowed a fine finish into the corner from 12 yards.
Archer was proving a constant menace and he played the decisive role as United took the lead in first half stoppage-time.
Taking aim from the edge of the area, Archer whipped a low strike off the far post and watched in delight as the ball cannoned back into the diving Pickford before looping over the line.
When Beto shot wastefully wide from a good position immediately after half-time, it looked like another afternoon to forget for Everton.
But Danjuma dragged them level in the 55th minute, the Dutch forward surging to meet Nathan Patterson's cross with a composed close-range finish.
With just seconds left in stoppage-time, Pickford made an astonishing double save to ensure Everton's losing run came to an end.
After pushing McBurnie's effort onto the bar, Pickford turned the striker's follow-up effort onto the post before breathing a sigh of relief as the final whistle sounded.
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