The draw extends the Blades' unbeaten run to six Premier League matches and lifts Manchester United up to ninth in the table.
Here, the PA news agency takes a look at the head-to-head battle between the two managers.
Harry Maguire captained Manchester United as the centre-back returned to his former club.
Scott McTominay's injury saw Phil Jones come in for his first Premier League start of the season.
Dean Henderson, on loan from Old Trafford, was replaced in the Sheffield United goal by Simon Moore, who not only made his top-flight debut but a first league appearance since May 2018.
Wilder's other alteration was also enforced as Phil Jagielka made his first league start since returning to Bramall Lane following an injury to the previously ever-present John Egan.
With Jagielka a like-for-like replacement for Egan, Wilder was able to deploy his usual 3-5-2 formation. The Blades attacked with gusto and in numbers while two banks of four defensively without the ball made it difficult for the visitors to penetrate.
His opposite number deployed a 3-4-3 formation with Daniel James, Marcus Rashford and Anthony Martial in a three-pronged attack. That became 4-3-3 when Jesse Lingard replaced Jones at half-time, with youngster Brandon Williams playing on the left side of a midfield three and scoring the visitors' opener in the 72nd minute.
Wilder made his first change after 68 minutes, with his side leading 2-0. Lys Mousset left the field to a standing ovation as club-record signing Oli McBurnie came on. Ten minutes later David McGoldrick was also applauded from the pitch as Billy Sharp entered the action, although by then the score was level at 2-2. And the Blades were behind when Wilder made his final change with seven minutes left, sending on striker Callum Robinson for defender Chris Basham. The hosts got their equaliser through McBurnie in the 90th minute.
Solskjaer decided to bring off Jones at half-time and introduce midfielder Lingard in a change of shape which ultimately paid off. Mason Greenwood replaced Andreas Pereira after 73 minutes and it was an inspired move by Solskjaer as the youngster equalised four minutes later. Axel Tuanzebe was brought on in the 85th minute as United tried to hang onto their lead, with Martial making way.
Wilder took up residence in his technical area and was animated from the off as he exhorted his players to build upon an encouraging start. He allowed himself a little celebration when Mousset added to John Fleck's opener to make it 2-0 but his head was in his hands after Rashford put the visitors in front. His mood changed dramatically, though, when McBurnie grabbed a late equaliser.
Solskjaer alternated between his seat on the bench and forays to the touchline, sharing the task of imparting instructions and encouragement to his players with assistant Mike Phelan. He celebrated wildly with his coaching staff after Rashford made it 3-2.