The result left the battling Blades still adrift in the Premiership drop zone, with their fans now hoping for a huge influx of cash for manager Neil Warnock to spend in January, which Magnusson is now expected to provide for the Hammers.
The biscuit magnate from Iceland is certainly not shy either, carrying his two-year-old grandson -also named Eggert - on to the pitch with him as he saluted the fans before the game to the inevitable tune of 'I'm Forever Blowing Bubbles'.
The Upton Park faithful had joined in with that raucously but the opening half an hour gave them little else to shout about until Mullins broke the deadlock with his close-range header.
Carlos Tevez, the Argentinian enigma brought to the club by the man who failed to win the takeover battle, Anglo-Iranian Kia Joorabchian, could have got the party started much earlier in fact when he spurned a fine fifth-minute chance.
Lee Bowyer had been the architect with a fine ball over the top of the Blades defence and Tevez's technique was just perfect to take it down and show the last defender his heels. Not so the finish however; that ended in the side-netting.
United hustled and bustled and might have take the lead themselves in the 21st minute when defender Claude Davis, who was finding Tevez a handful, arrived in the Hammers' six-yard box to meet Keith Gillespie's corner.
No home defender challenged and West Ham were grateful that Paul Konchesky was able to shepherd his downward header round the back post for a corner which this time was dealt with.
Blades midfielder Nick Montgomery picked up a 29th-minute booking for a late challenge on Nigel Reo-Coker as West Ham finally took control.
Rob Kozluk did well to stop Bowyer opening the scoring with a timely block after Bobby Zamora, having shaken Davis off down the right, had set him up.
But that intervention, at the expense of a corner, merely delayed affairs as Mullins nodded home from Matthew Etherington's corner, which had been flicked on at the near post by Anton Ferdinand.
The Blades did manage to put the ball in the back of the net before the break but referee Mike Riley had already blown his whistle for a push by Davis.
United were more of a threat after the break with the teenager Nicky Law firing a good chance across goal and wide but it was not until the hour mark, when Christian Nade and Steve Kabba had been sent on to help Rob Hulse up front that the chances started arriving regularly.
Danny Gabbidon had to stop Nade going clear within seconds and was carried off in agony soon afterwards. That rocked the Hammers but Nade had already squandered what would turn out to be the Blades' best opportunity of the afternoon.
Gillespie fired in a cross from the right which found the substitute unmarked in front of goal but somehow he managed to stab wide from five yards.
Teenager Nicky Law saw a shot deflected inches wide before Davis' header forced a diving stop from England goalkeeper Robert Green.
The Hammers were having to hang on and manager Alan Pardew sent on forwards Teddy Sheringham and Marlon Harewood, who went on to miss a great chance to make it 2-0 in stoppage time, for the closing period.
But hang on they did with Kozluk had a strike ruled out for a foul on Green at the death.