But, after numerous scares and countless anxious moments, Chris Wilder’s side secured the result which kept their hopes of reaching the Championship play-offs alive.
Although the scoreline suggests otherwise, this was far from a routine victory.
Indeed, until substitute David Brooks stretched United’s advantage midway through the second period, Albion seemed destined to take something from the game thanks in no small part to their opponents’ uncharacteristically fractious approach.
Wilder is too experienced, too long in the tooth, to argue that performances rather than points are the most valuable currency during the closing stages of a promotion challenge and he will be encouraged by their persistence.
But he will also be aware that, despite climbing to seventh in the table, United could suffer the consequences if their pedestrian first-half display is repeated against the likes of Middlesbrough, Preston North End or Cardiff City who all travel to South Yorkshire between now and May.
Indeed, it was not until Mark Duffy’s perfectly weighted pass giftwrapped Brooks’ goal that United, who had earlier seen Enda Stevens open his account for the club, finally discovered their rhythm.
As well as being pivotal - “These two home games...will go a long way towards deciding where we finish,” Wilder had admitted in his programme notes - last night’s fixture was poignant too.
Nigel Clough, who led United into the semi-finals of both the FA and EFL Cups before being sacked in 2015, is still piqued by the manner of his departure.
Meanwhile, sat a couple of yards over his shoulder in the GAC Stand, Paul Coutts reflected on the collision which ended his season when the two clubs met at the Pirelli Stadium earlier this term.
Replacing the midfielder, whose absence coincided with United’s worst sequence of results so far this term, has proved a tougher ask than even Wilder first feared.
But Mark Duffy’s return from the calf injury which forced him to miss last weekend’s draw with Ipswich Town ensured the hosts, as he later demonstrated in devastating fashion, were equipped with a creative edge.
Despite their respective league positions, Albion have enjoyed plenty of success against clubs towards the top of the rankings.
James Wilson, recalled to the starting eleven at the expense of Ched Evans, received an early reminder this fixture would be anything but routine in the shape of a boot to the face as he challenged for possession near the centre circle.
Retribution came when Lee Evans, sold dreadfully short by John Fleck, flew into a tackle with Jamie Allen which saw the youngster join Albion’s long list of walking wounded.
Nevertheless, it was the visitors who carved the evening’s first serious chance when Lloyd Dyer, peeling away from his marker, forced Jamal Blackman into a fine reaction save.
Whether or not it was Albion’s tactics or merely a symptom of the gruelling fixture schedule, United’s work lacked precision and purpose until Stevens’ 29th minute intervention.
Fleck did his best to rectify the situation by smashing a cross into Stephen Bywater’s area which simply begged to be turned home but neither Billy Sharp nor Wilson could deliver the vital touch.
But the Irishman, combining with his fellow wing-back George Baldock, had no such problem when he adjusted his body and turned home from close-range.
If United thought that would mark the end of Albion’s resistance, they were sorely mistaken.
Indeed, as the interval approached, it was the visitors who took charge.
Blackman, who later denied Dyer for a second time, bravely smothered the ball at Darren Bent’s feet following some powder-puff defending before Tom Flanagan’s cross was inches away from being turned home by Hope Akpan.
With Blackman wrongfooted, Albion would have been level if his shot had been on target.
United began the second period with a drive which suggested some choice words had been exchanged in their dressing room during the interval.
Baldock, ghosting into space at the far post, was found by Fleck but failed to make a clean connection.
Then, moments later, Sharp clattered into the upright as he stretched to meet an inviting Wilson cross.
When Bywater thwarted Duffy with an excellent one-handed block, it appeared as if United had finally taken charge.
Instead, the momentum swung back to Albion and nerves inside the stadium became increasingly frayed until Brooks, profiting from Duffy’s piece of magic, put the contest beyond reach.
Only the width of the crossbar and Bywater’s reactions prevented Evans and Fleck from stretching United’s lead as they moved to within two points of sixth-placed Middlesbrough.
Sheffield United: Blackman, O’Connell, Stearman, Basham, Stevens, Baldock, L Evans, Fleck, Duffy (Lundstram 85), Wilson (Brooks 61), Sharp (Donaldson 90). Not used: Moore, Leonard, Lafferty, Holmes.
Burton Albion: Bywater, Allen (Sbarra 12), McFadzean, Dyer, Flanagan (Buxton 82), McCrory, Naylor, Sordell, Davenport, Akpan, Bent (Varney 82). Not used: Campbell, Barker, Boyce.
Referee: Scott Duncan (Northumberland).