The victory sends the Hammers to their first Cup semi-final in 20 years.
Nsofor, who notched up his first goal in English football only last weekend, set up all four goals to snap United's 29-match unbeaten streak in all competitions.
Playing under snowy conditions, Obinna had the ball inside the Man United net but the effort was correctly chalked off as it deflected in off Jonathan Spector, who was clearly offside.
But both players were to combine to rectify the error with their next attack.
Obinna, receiving a Carlton Cole pass, dinked a clever cross to Spector, allowing the former United man to steal in front of Darren Fletcher, and head beyond Tomasz Kuszcak.
Incredibly, a second was to follow with United looking uncharacteristically disjointed. Again, it was Spector, who has never scored before in his 97 appearances in English football. And again, it was Obinna with the assist.
Nsofor continued to be a threat, and he was on the end of a quick free-kick by Luis Boa Morte and, with Rafael struggling in the conditions, dropped a cross in the box which Cole headed past the napping Jonny Evans.
With his confidence sky high, the Nigerian turned Rafael this way and that before sending in a low cross. Cole again was on hand to twist his way past Evans and stroke his shot beyond Kuszczak. With a Premier League victory under their belt over the weekend, West Ham, it would seem, have turned the corner.