Cameron Archer scored on his league debut for the club to cancel out Abdoulaye Doucoure's first-half effort, before Jordan Pickford's own goal gave the hosts the lead at half time.
However, Arnaut Danjuma scored for the second successive game to earn a point for the Toffees and ensure that both teams got off the mark at the fourth time of asking.
Sheff Utd started the match with their tails up after pushing the reigning champions Manchester City close last weekend, but it was the visitors who struck first after 14 minutes when Doucoure bundled the ball home from a corner, which had been won after Beto's effort from the edge of the area deflected wide.
Amadou Onana climbed highest to make first contact from James Garner's out-swinging corner, with the ball landing in the direction of Doucoure, who prodded straight at Wes Foderingham. However, Sheff Utd's goalkeeper could only divert the ball back into the path of Doucoure, who tapped into an empty net to send the away fans into bedlam.
The hosts' heads did not drop, though, with a nice switch of play from Oliver Norwood helping create space for George Baldock and Gustavo Hamer to combine, but the former Coventry City midfielder's strike was pushed wide by Pickford.
However, England's first-choice goalkeeper could do nothing to prevent the Blades from getting themselves on level terms when Archer - making his Premier League debut for the club following his £18m move from Aston Villa - expertly fired into the corner from 12 yards after 33 minutes. Archer's strike partner Oliver McBurnie had brilliantly controlled the ball from Baldock's cross, before laying it nicely into his teammate's path.
Archer was hugely involved in helping his new side head into the half-time interval with a one-goal cushion as he cut inside from the left before firing an effort against the post, with the ball rebounding off Pickford's back before nestling into the back of the net.
As expected, with the onus on them to find an equaliser, Everton played with more initiative after the break, with the lively Danjuma firing into the wall from a free kick after forcing Anel Ahmedhodzic to take a yellow card by bringing him down.
The Netherlands international had regularly found himself in plenty of time and space down the left flank throughout the first half, but his end product had been lacking until firing the visitors level in the 53rd minute.
Nathan Patterson's fine cross from the right flank found him at the back post, and he did not disappoint by meeting the ball on the half-volley to bring Everton level.
With both sides cancelling each other out for large spells thereafter, McBurnie almost snatched all three points in the last minute of injury-time, but Pickford's sharp reactions were on full display to deny the towering target man twice.
Both teams will be relieved to get themselves off the mark in terms of points heading into the international break, with the respective managers likely to be buoyed by the attacking vibrancy on display from both sets of players.