After 111 years under East End ownership, baking baron Magnusson's £108million midweek takeover had led Alan Pardew to concede that his 16th-placed side needed to get back to winning ways under the new regime.
And having spent all that dough, the Icelandic investor, who was given a traditional Cockney welcome, saw West Ham fittingly rise to the task.
Neil Warnock's strugglers - in 18th spot - had only managed a trio of league and cup victories all season and, after last Saturday's defeat against table-topping Manchester United, the Blades boss made just one change as Nick Montgomery came in for Colin Kazim-Richards.
Following last week's narrow reverse at fortress Stamford Bridge, West Ham, with only three wins, too, had retained Carlos Tevez in an unchanged line-up.
Certainly, the Argentinian had impressed against Chelsea and within four minutes he found himself racing on to Lee Bowyer's quickly-taken free-kick and drilling a low, angled shot into the side-netting.
But the organised Yorkshiremen quickly re-grouped and, at the other end, Nicky Law sent a rising, angled shot flying behind on the quarter-hour mark, before Paul Konchesky scrambled Claude Davis' downward header away from the base of his right-hand post.
With neither side able to find the killer pass, the game looked set to degenerate into a midfield stalemate and Montgomery's caution for felling the fleeing Nigel Reo-Coker only served to heighten the Anglo-Icelandic anxiety.
Ten minutes before the interval, the breaking Bobby Zamora's unselfish pass into the danger zone looked to have done the trick, only for the sliding Robert Kozluk to divert Bowyer's goalbound shot wide.
That was only a temporary reprieve, though, for when Matthew Etherington floated over the consequent corner, Anton Ferdinand's flick-on was met by the in-rushing Mullins, who powerfully nodded Hammers' into an interval lead with his second goal of the campaign.
If West Ham had edged the first period, then Sheffield United soon gained the upper hand after the break as Law, Montgomery and Alan Quinn each tried their luck with early efforts.
But as the Blades struggled to find any true cutting edge, there were yellow cards for the dissenting duo of Rob Hulse and Keith Gillespie, whose frustration was further heightened by Danny Gabbidon's brilliant saving tackle on substitute Christian Nade.
Shortly afterwards, the Hammers' Welsh defender was stretchered away with a hamstring injury and as the East Enders re-grouped, Nade toe-poked Gillespie's inch-perfect cross wide from just eight yards, before Law drilled an angled effort beyond the far post.
West Ham offered little in reply and not even the introduction of Teddy Sheringham and Marlon Harewood for the toiling Tevez - who instantly stormed away from Upton Park - and Zamora could inject any impetus.
Instead the fresh legs of the Sheffield substitute trio - Nade, Steve Kabba and Kazim-Richards - set up a frantic finale in which Mikele Leigertwood went close and Kozluk had a last-gasp tap-in ruled out to the dismay of Warnock and the joy of Mr Magnusson.