Nuno Espirito Santo's side look set for a big battle for a Europa League place now, too, with the Blades only a point off them thanks to Egan nodding home deep into added-on time.
Another disappointing result - after being downed 2-0 by Arsenal last weekend - and the standard needs raising against Everton on Sunday.
This was all far too negative from Wolves. For the second straight game, they only registered one shot on target.
Assessing things a whole, sixth is still a nice place to be - but rather than staring up with excitement, Nuno's men are now anxiously looking over the shoulders.
This game rarely set pulses racing for those watching at home.
Credit must go to United - they are a very well-drilled outfit who fully deserve to be where they are in the table.
They were not troubled in the second half, though. The first half saw Wolves provide some bright moments - but it all petered out after the break until United piled on the pressure and, eventually, won it through Egan.
Wolves, make no mistake, are still in a good position, but they must show more ambition.
Ruben Vinagre was left on the bench, as was Daniel Podence. A spark was needed but they were punished for being too conservative.
Looking to respond to Saturday's disappointing defeat to Arsenal, Nuno put Wolves back in the more ambitious 3-4-3 set-up.
Diogo Jota returned to the starting XI as Leander Dendoncker dropped to the bench. Also included among the substitutes was Podence - given league clearance to return to the fold following his trip back to Portugal for personal reasons.
Pedro Neto missed out altogether, though, because of the knock he picked up against the Gunners while veteran Billy Sharp was up front for the hosts.
Wolves, of course, do not tend to start matches all that quickly. They went into the encounter with the worst first-half record in the top flight, but with Jota and Adama Traore out wide, there was every chance of bucking the trend. As it turned out, the streak continued - but not due to a lack of effort.
Nuno had a heart-in-mouth moment mere seconds into the clash.
Traore - having already dislocated his right shoulder three times this campaign - went down clutching it when trying to shield the ball. Thankfully, with a bit of treatment, he was able to continue, even though Nuno confirmed afterwards it was another dislocation.
Wolves were a little tentative for a few minutes after that early scare, and perhaps understandably so.
But around the quarter-hour mark, they began to impose themselves in the final third.
Traore beat the offside trap and eventually hooked it over to Raul Jimenez, who was only denied by a heavy touch. Jota then tried to cut the ball back to the Mexican, with the low cross bouncing out for a corner. Not quite, but the intent was certainly there.
Jota looked lively, linking up play with a series of clever touches between the lines. Wolves, after all, are a better team with him in it.
The Portuguese helped them almost break the deadlock, too. Latching onto a lovely flick from Jimenez, he drew a cynical foul from Jack O'Connell - booked - on the edge of the box.
Ruben Neves licked his lips with excitement as he stepped up to take the free-kick from 20 yards, and saw his fantastic, floated effort smack the bar. Dean Henderson cut a relieved figure.
Neves would send another strike narrowly high and wide from trademark before the half-time whistle blew.
The goalless first-half run now stands at six league matches, but there was room for encouragement as - aside from an Oli McBurne header, which went straight at Rui Patricio - they were the ones asking the questions. It was an improvement on the opening 45 against Arsenal, for sure.
However, the Blades made the much sharper start to the second period. Traore and Jota both stumbled on the increasingly-slick pitch, thanks to the persistent rain, and Wolves became more careless in possession.
Sharp stuck the ball into the back of the net for the growing-in-confidence United with a neat turn and finish - but, fortunately for Nuno & Co, he was correctly flagged offside. A teasing cross from Enda Stevens then just evaded Sharp, on the stretch.
It had gone flat for Wolves, with Traore off the boil. They created nothing going forward, apart from a counter-attack that fizzled out when Jota could not connect properly with his head. Stevens hacked down Traore during that and became the 34th cautioned for fouling the Spaniard this season.
Ultimately, though, things needed freshening up. A magical moment was needed.
But Nuno, surprisingly, waited until the 79th minute to make a switch, and it was Dendoncker who entered the fray for Traore in what seemed a concerted effort to hold onto a point, reverting to 3-5-2.
Instead, the home side won it as centre-half Egan came up with a towering header. A vastly frustrating evening for Wolves.
Sheff U (3-5-2): Henderson; Basham, Egan, O'Connell; Baldock, Berge, Norwood, Osborn, Stevens; McBurnie (Mousset, 49), Sharp (Zivkovic, 71)
Subs not used: Moore (gk), Jagielka, Robinson, Freeman, Rodwell, Lundstram
Goals: Egan (90+4)
Wolves (3-4-3): Patricio; Boly, Coady, Saiss; Doherty, Moutinho, Neves, Otto; Traore (Dendoncker, 79), Jimenez, Jota
Subs not used: Ruddy (gk), Kilman, Vinagre, Buur, Jordao, Gibbs-White, Podence, Campana
Referee: Martin Atkinson (West Yorkshire)