As Andoni Iraola's side failed to make any of their numerous chances count - one of which saw the luckless Dominic Solanke slip and fire a penalty over the bar - the Blades punished them through Gustavo Hamer and Jack Robinson's efforts either side of the half-time whistle.
Substitute Dango Ouattara cut the deficit in half with 16 minutes of normal time to go, though, and just as the clock ticked past the 90, Getafe loanee Unal opened his Bournemouth account to cruelly deny their visitors an elusive win.
With Solanke cleared to start amid his ongoing knee issue, Chris Mepham for the stricken Marcos Senesi was the only change on the hosts' end, while Ben Osborn, George Baldock, Ben Brereton Diaz and Ollie Arblaster - making his first Premier League start - were introduced into the visitors' XI.
Despite their status as the Premier League's worst performing away side - in terms of points, wins, goals scored and goals conceded - Chris Wilder's men had an early sighter in the fifth minute through one of their quartet of changes, but Brereton Diaz's drive was straight down Neto's throat.
Iraola's men quickly began to turn the screw, though, and they won a penalty in the 12th minute, when Tom Davies needlessly challenged Solanke from behind inside the box.
The Cherries' top scorer dusted himself down and stared Ivo Grbic down, but to the agony and astonishment of the home crowd, Solanke slipped as he struck his spot kick and ballooned his effort several metres over the crossbar.
Even if the striker's penalty had rippled the net, a double touch would have rendered it meaningless anyway, but Bournemouth continued to advance in droves, and Antoine Semenyo tested the gloves of Grbic twice in the 20th and 23rd minute.
It would seemingly only be a matter of time before the deadlock was broken, and the first goal of the game would arrive with 27 minutes on the board, but it would surprisingly be the men in yellow celebrating.
Strong centre-forward play from Oli McBurnie saw the striker free Jayden Bogle down the right, and Neto could only parry his low strike back into the danger area, where Hamer arrived to fire home a shock opener for the Blades; the Bournemouth number one got a hand to his strike to no avail.
Undeservedly trailing, Iraola's men continued to carve out openings at will towards the end of the half, but Sheffield United went into the break with their one-goal advantage intact, leading Iraola to make an attacking change at the break as Ouattara replaced Milos Kerkez.
Normal service resumed in the second 45, as the hosts attacked the Sheffield United backline at every opportunity and had a couple of penalty appeals waved away, first for Vinicius Souza's 58th-minute challenge on Ryan Christie before the VAR room saw nothing wrong with Hamer's 61st-minute tackle on Justin Kluivert.
However, Bournemouth's hard work in the final third was undone by a goalkeeping calamity in the 64th minute, as Neto punched Hamer's delivery onto Solanke's back, and Robinson reacted quickest to hook the rebound over the line before Neto could claw the ball away.
The Cherries thought that they had cut the deficit in half just two moments later, as Robinson's attempted clearance struck Solanke's midriff and set the striker up to bundle home, but upon a VAR review, a handball offence was spotted; the Englishman knew little about it, but his goal was chalked off to exacerbate his nightmare afternoon.
Iraola's frustrated crop would belatedly breach the Blades' backline in the 74th minute, as Ouattara made a darting run to meet Christie's corner at the near post, and the Bournemouth substitute produced a majestic leap before directing a fierce header over Grbic.
An almost identical scenario led to a Bournemouth equaliser in the 82nd minute, but this time - as Grbic came out to meet the delivery from the left and missed - Ouattara's header cleared the top of the crossbar.
Eight minutes of injury time rekindled the Cherries' hope, though, and in the first additional minute, two substitutes would combine to complete the second-half turnaround, as Sheffield United's set-piece frailties proved costly.
The visitors failed to properly clear a corner, and as the ball was hooked back into the mixer, Ouattara headed the ball on for Unal, who out-muscled his man at the back post and side-footed home his first Bournemouth goal on the volley.
The Cherries huffed and puffed for the winning goal - Luis Sinisterra and Philip Billing both had shots off target in the final moments - but they would settle for the point which keeps them 13th in the table, while Wilder's men have at least climbed off the bottom despite their late collapse.
Bournemouth have one more engagement before the upcoming international break, hosting Luton Town on Wednesday evening, while Sheffield United - who hold a one-point lead over new basement boys Burnley - return to action in three weeks' time at home to Fulham.