The Blades suffered a dismal relegation from the Premier League in the 2023-24 season but have enjoyed an excellent campaign this time around. They led the league going into the final seven games only for a run of four defeats from five in April to consign them to third place and the play-offs.
Much of the damage in this semi-final was done in the first leg when Rob Dickie's red card gave them a penalty kick that Burrows scored before the break and subs Andre Brooks and O'Hare added to.
City, needing to achieve something no team has done in play-off history and recover from losing the first leg at home by a margin of three goals, made an encouraging start but Scott Twine miscued their best opening.
United grew into it and got the first goal of the night when Moore was able to hold off a defender and head in from a corner kick with goalkeeper Max O'Leary unable to get to the ball first.
That ended any incredibly faint hopes that Liam Manning's side had and the Blades' second of the night summed up their semi-final.
First they coughed up possession on the halfway line to allow the home side to break and win a corner kick, then they failed to pick up Hamer on the edge of the area from the ensuing set-piece and his effort took a huge deflection to go in the far corner.
O'Hare, a beaten play-off finalist with Coventry two years ago, scored his second of the tie when Burrows added to his two assists of the night by stepping over Tyrese Campbell's low cross into the penalty area to allow the midfielder to convert first time.
In emotional scenes after the game, United players and staff celebrated with a Greece flag in honour of former Blades full-back George Baldock, who died at the age of 31 in October.
The South Yorkshire side have failed in all nine of their previous play-off campaigns and have failed to score in all four of their play-off finals to date. They will look to end what is the worst play-off record in the EFL a week on Saturday.