The best chance in the first half of this battle between two sides with European ambitions fell to Wolves' Ruben Neves, whose free-kick hit the crossbar.
Sheff Utd's George Baldock headed Richairo Zivkovic's cross wide and Ben Osborn's effort was turned behind, and the home team's persistence paid off in the final few minutes.
Defender Egan rose highest to head in Oliver Norwood's corner and take Chris Wilder's team to 51 points in seventh position.
Back-to-back losses for Wolves leave them three points behind Manchester United, who have a match in hand.
Sheff Utd: 11 Jul v Chelsea (H)
Wolves: 12 Jul v Everton (H)