Sharp fired in from the spot after Ademola Lookman’s excellent solo effort had put the visitors ahead at Bramall Lane.
The Fulham winger, making his first start for the club, skipped beyond two players before firing left-footed past Blades goalkeeper Aaron Ramsdale
Aleksandar Mitrovic had earlier smacked a penalty against the crossbar after defender Jack Robinson handled.
And it was Mitrovic who then fouled Robinson inside the Fulham box, with the decision awarded by the video assistant referee.
The draw means both teams have their first point of the season, after losing their opening four games.
Lookman shines as Mitrovic despairs
Scott Parker has packed his Fulham squad with young talents with something to prove, and handed first starts to three players who have represented England at different levels.
Ruben Loftus-Cheek, on loan from Chelsea, and defender Tosin Adarabioyo, who has signed from Manchester City, made their debuts, but it was Lookman who almost stole the show.
The 22-year-old has had an enigmatic career so far, failing to hold down a place at Everton and then RB Leipzig.
He impressed Parker with a lively cameo off the bench against Wolves last time out and took his first Premier League goal in three years superbly on Sunday.
The winger cut inside Sander Berge before forcing his way into the penalty box and driving a smart finish beyond Ramsdale with 13 minutes to go.
Mitrovic had missed from the spot earlier in the half and then headed wide late on from a wicked Lookman delivery.
Between those chances, the forward gave away a penalty at the other end when he caught Sheffield United defender Robinson – becoming the first player to both miss and give away a spot-kick in the same Premier League game since Mikel Arteta for Arsenal in November 2012.
The result was enough to lift Fulham off the bottom of the Premier League table, to 19th.
Experienced Sharp rescues Blades
Chris Wilder has worked wonders at Sheffield United, guiding the Blades to their best league finish in 45 years on their return to the Premier League last season.
But their second term in the top flight is proving more difficult so far.
Oliver McBurnie should arguably have opened the scoring in the first half, but Fulham goalkeeper Alphonse Areola reacted quickly to push his close-range header away from goal.
McBurnie also flashed a header wide from Chris Basham’s cross, and Areola then saved comfortably from an Enda Stevens effort.
The Blades thought they had taken the lead when Fulham defender Adarabioyo turned the ball into his own net, but Berge was offside when sliding the ball across goal.
Wilder brought on 20-year-old club record signing Rhian Brewster for his debut with just under half an hour to go, but it was experienced striker Sharp who salvaged the hosts a point.
The 34-year-old made no mistake with his penalty to score Sheffield United’s first goal at Bramall Lane in 368 minutes of Premier League football.
The draw lifts Wilder’s side out of the relegation zone on goal difference, but things are not getting any easier for them with their next three fixtures against Liverpool, Manchester City and Chelsea.
‘The team is going in the right direction’
Sheffield United boss Chris Wilder told Match of the Day: “We were a little bit more like our old selves in the first half and had control of the game, but we were not like our old selves in the second half. They dominated the second half.
“It is a bit of a battle at the moment, a bit of a grind, but delighted to get something out of the game.
“I am not going to complain about the (penalty) decision because we have talked about some that have gone against us this season, so we have had a touch of luck today. We will take that and I think it is due.”
Fulham boss Scott Parker told Match of the Day: “I thought in the second half we were exceptional. In the first half, we came with a bit of plan, we needed to tinker things at half-time and the players responded very well.
“The second half, I saw a dynamic team, a team that could create chances and I was pleased overall in that sense. There is still plenty of learning and plenty of improvement to do – a young team with young players coming in and making their debuts.
“The team is going in the right direction, not just today. The last performance as well was very pleasing. From where we were in the first three games where we had our struggles we appear to be coming out of that and pushing in the right direction.”
The stats – rare Premier League draw for Fulham
- Sheffield United have failed to win any of their opening five league games to a season for the first time since 2006-07, when they were relegated from the Premier League.
- This was Fulham’s first draw in 15 Premier League matches under manager Scott Parker (won three, lost 11), and their first in 24 matches in the competition since a 1-1 draw with Wolves in December 2018.
- Fulham have missed three of their past six Premier League penalties, as many as in the previous 34 they had taken in the competition.
- Fulham’s Ademola Lookman scored his first goal in 37 Premier League appearances; his only previous goal in the competition came on his debut against Manchester City for Everton in January 2017.
Sheffield United visit Liverpool on Saturday (kick-off 20:00 BST), while Fulham host Crystal Palace earlier that day (15:00).