For the £4m summer arrival from Brentford – who fired the Blades in front with a first-half header after goalline technology ruled the ball had crossed the line – should have grabbed a second-half winner, but missed with the goal at his mercy in front of the Kop.
It looked set to be a costly error, and two dropped points, until captain Sharp’s later intervention with his third goal in as many games.
Egan said: “In the second half, I had a chance in front of the Kop, and it would have been brilliant to get the second, but, thankfully, that man Billy Sharp popped up in the ‘93rd minute’ and we managed to get a big win.
“As a team, we really dug deep for each other and we could see everyone was fighting for each other. That’s huge.
“Three in three now for Billy. He’s a brilliant goalscorer and any time he’s on the pitch and gets a chance, we’re in business. He just sniffs them out.
“My goal? No doubt about it, I knew straight away it was over the line. When I headed it, I was on the angle and saw it go over the line so I was just looking at the ref, waiting for him to blow the whistle really. We’ve got the technology now so no doubt about it.
“I probably should have scored a second, too, but won’t talk too much about that one!”
The Irish defender’s goal came in the ninth minute, his looping header from Ollie Norwood’s corner beating Tim Krull.
In spite of a last-ditch clearance from Tom Trybull, referee Darren Bond used goal-line technology to judge the ball had crossed the line.
But the Canaries were level when Jordan Rhodes – the on-loan striker from Sheffield Wednesday – celebrated his return to the Steel City with a poacher’s goal.
The striker had earlier missed with a header from eight yards out, but smashed in a close-range finish after impressive work down the flank from Teemu Pukki and Onel Hernandez.
Rhodes – Wednesday’s own club record signing at £8m – was jettisoned out of Hillsborough to help reduce the Owls’ inflated wage bill, not fitting into the style of play under Jos Luhukay.
But he has scored twice in his first three league starts for Norwich, and Egan enjoyed his battle with the striker.
“That’s the type of striker Rhodes is. He comes alive in the box and we had to be wary for the whole game,” said 25-year-old Egan. “He took his goal well and apart from that didn’t really touch the ball.
“That’s the type of player he is – one chance, one goal. He’s done it all his career and you have to take your hat off to him sometimes; he’s a clinical striker. You have to be on your toes the whole game.”
Both sides had good chances to nudge themselves in front. Leon Clarke fired straight at Krull, who later palmed away Egan’s athletic volley, and Ryan Leonard’s header was off target.
The Blades were indebted to goalkeeper Dean Henderson, who thwarted Moritz Leitner in a one-one-one, despite the Canaries man starting from an offside position.
United then had a double let-off, as Rhodes headed against the inside of the post before Pukki scooped the ball over from four yards out.
Then came Egan’s glorious chance to grab the winner. Norwood – making his league debut – produced another pinpoint cross, and the defender met the ball with a firm header only to see it flash wide of Krull’s right-hand post.
It looked like proving a costly miss, but deep into the three minutes of stoppage time, Chris Basham’s cross was headed back across goal by substitute David McGoldrick and Sharp grabbed the winner with a close-range finish. The goal sparked huge celebrations in front of the Kop – even fist-pumping manager Chris Wilder raced down the touchline to join in – as United secured back-to-back Championship wins.
Egan said: “The scenes at the end were brilliant, ‘93rd minute’ winner doesn’t get much better than that – best way to win a game.
“The feeling with the fans was brilliant. They stayed behind us all game and drove us over the line which is all you can ask for.
“It was a tough game. I thought we deserved the three points overall. Both teams had a go but we created better chances so we deserved it.”
Egan reserved special praise for midfielder Norwood, the stand-out performer on the pitch on his league bow for United.
Signed on loan from Brighton, Norwood impressed on debut in the League Cup exit to Hull City in midweek, and the 27-year-old was excellent on Saturday, providing a conveyer belt of quality crosses. “His delivery is on the money,” explained Egan. “We had a few chances from set pieces, consistent, too, and we always know you have a chance to attack the ball.
“You could see his quality on Tuesday against Hull and again today. We had a few chances from set pieces – they can sometimes decide a game and, luckily, we got one today.”
Sheffield United: Henderson, Stevens, Fleck, O’Connell, Basham, Clarke, Sharp (Stearman 90), Woodburn (Leonard 45) Egan, Norwood, Freeman (McGoldrick 61). Unused substitutes: Lundstram, Stearman, Lafferty, Moore, Lavery.
Norwich City: Krul, Marshall, Leitner (Stiepermann 78), Rhodes, Lewis, Klose, Trybull, Pukki (Srbeny 87), Hernandez, Tettey (Godfrey 90) Hanley. Unused substitutes: Zimmermann, McGovern, Thompson, Aarons.
Referee: D Bond (Lancs).