The League One side went in front against the run of play in the 28th minute at Goodison Park through Richard Cresswell, but the same player diverted the ball into his own net moments later after Steve Simonsen had got a palm to a strike by Jack Rodwell.
The Toffees then took control of proceedings, with goals from Victor Anichebe and Mikel Arteta swiftly turning the game on its head in the space of five minutes before the break.
The result will come as a welcome boost for Everton fans after the Merseysiders' season had got under way in disappointing fashion with Saturday's 1-0 Barclays Premier League defeat at home to QPR.
Manager David Moyes made five changes from that game, with Jan Mucha, Tony Hibbert, Arteta, Marouane Fellaini and Anichebe all coming into the starting line-up.
Teenage midfielder Ross Barkley kept his place after making an impressive senior debut in the match at the weekend, while former Everton goalkeeper Simonsen was named in the visitors' first XI.
Fellaini brought a save out of Simonsen in the fifth minute with a low shot from the edge of the area before Barkley, showing more of the confidence that had been so evident in his previous display, rampaged forward and fired an effort wide.
Arteta had a strike blocked at the end of a good Everton move and Anichebe put the ball the wrong side of the post after turning in the box.
Further attempts followed from Barkley and Fellaini, with the only respite for the Blades being a shot well off-target by Harry Maguire.
Arteta was presented with the best chance yet to open the scoring in the 20th minute but made a hash of it from close range.
Fellaini muscled through a clutch of defenders to unleash an effort which deflected wide, and Barkley scooped the ball over the bar.
United had offered virtually nothing in attack, but just prior to the half-hour mark they suddenly snatched the lead as Cresswell slid in to meet Stephen Quinn's cross and prod it past Mucha.
The advantage lasted a mere three minutes, Simonsen parrying Rodwell's strike at the other end and the unfortunate Cresswell getting the final touch as the ball bobbled over the line.
Anichebe then put the Toffees ahead, connecting with Barkley's drive across the area to sweep home.
And Everton had another three minutes before the interval as the ball broke to Arteta, who stroked it beyond Simonsen from just outside the box.
Quinn called Mucha into action with a shot six minutes after the restart, but normal service was soon resumed as Simonsen saved efforts from Barkley and Leon Osman.
Everton continued to press, with Anichebe volleying wide before Fellaini and Arteta sent attempts too high.
Moyes then added fresh legs to his forward line, throwing on Louis Saha in place of Rodwell.
Blades substitute Daniel Bogdanovic tried something ambitious which sailed over before Osman nearly added a fourth for the hosts, rifling the ball against the post with 15 minutes of normal time remaining.
The midfielder was then replaced by Phil Neville, while another eye-catching performance for Barkley came to an end as he was substituted for Jose Baxter.
In the last 10 minutes Matthew Lowton directed a header into the arms of Mucha, who also tipped over a shot by Maguire.
It was Everton who almost finished with a flourish, though, Anichebe finding himself through on goal only for Lecsinel Jean-Francois to make a last-ditch tackle.