The centre-half drilled the ball home from the edge of the six-yard box in the 80th minute after substitute Billy Sharp had nodded on Ben Osborn's cross.
James Tarkowski had put Burnley ahead just before half-time when he prodded in the ball after Jay Rodriguez's flick-on from Dwight McNeil's free-kick.
McNeil could have doubled the hosts' lead but the winger shot a great opportunity wide three minutes before Egan's equaliser.
Burnley move above Tottenham Hotspur into ninth place on 46 points, two points behind Sheffield United, who remain in eighth.
Burnley: 8 Jul v West Ham (A)
Sheff Utd: 8 Jul v Wolves (H)