Spector had never scored during his six seasons in the Premier League but the USA star struck twice to begin a rout that sent Hammers fans delirious.
Further damage was done by Cole after the interval with the England marksman pouncing twice to bring feeble United crashing back down to earth after their 7-1 drubbing of Blackburn.
The defeat also ended the Red Devils' 29-match unbeaten run in all competitions.
Spector and Cole were brilliantly supported by Victor Obinna with the Nigerian forward, on loan from Inter Milan, having a hand in each goal.
Sir Alex Ferguson made 10 changes to the side that hammered Blackburn last weekend with only Anderson remaining, but the manner of this defeat will still prove difficult to stomach.
Experience was supplied by the likes of Ryan Giggs, Darren Fletcher and John O'Shea but they also wilted in the face of West Ham's onslaught.
Those who had braved the arctic conditions at a near-sold out Upton Park enjoyed a night of high drama, the benefit of which could be felt in the Hammers' Premier League survival bid.
Manager Avram Grant has proved something of a cup specialist but while he will be delighted to see his side through to the semi-finals, he will be just as pleased at having strung together successive victories that could inspire a climb up the table
Wigan were dispatched 3-1 last weekend but Grant made six changes with captain Scott Parker on the bench despite suffering from a chest infection.
There was little sign of the carnage to come early on when the holders were denied the lead by the woodwork, with Robert Green showing lightning reflexes to push Gabriel Obertan's shot on to the post.
Anderson expertly picked out Giggs but the Welsh winger's delivery into the box was poor and the ball looped away harmlessly.
Controversy reigned in the 16th minute when West Ham scored through Obinna - only to have the goal ruled out for offside.
Obinna's shot deflected off Spector and the linesman had made the correct decision, but the Hammers were incensed.
Players queued up to demonstrate with referee Mark Clattenburg, who then warned Grant after the Israeli hurled the ball away in disgust after it had gone out of bounds.
Any sense of injustice was soon remedied, however, when Spector nodded West Ham ahead.
Obinna supplied an exquisite chip that Spector, having crept inside Fletcher, headed into the net.
Emboldened by their goal, West Ham attacked with greater conviction but were almost caught by a United counter that ended with a tame header from Obertan.
Cole almost got the better of Fabio before James Tomkins angled a corner wide and in the 37th minute the home side's ascendancy was rewarded once more.
The hapless Fabio was involved as his attempted block of Obinna's shot instead fell into the path of Spector who blasted home from close range.
A header from Bebe was easily saved by Green but most of the action continued to take place at the other end.
A dreadful blunder from Green, who until now had looked rock-solid, almost let United in with his clearance hitting the back of Federico Macheda.
The ball deflected into the path of Javier Hernandez but reinforcements arrived for the Hammers to clear the danger in the nick of time.
United were pouring forward as the expected fightback began in earnest, only to be halted in their tracks in the 55th minute when they were caught on the break.
The superb Obinna was the supplier with his inch-perfect cross met by Cole, who headed the ball home.
There was still more to come from West Ham, however, as little over 10 minutes later Cole slammed home his second after being set up by Obinna.
Zavon Hines had a strong penalty shout rejected by Clattenburg after being shoved in the back by O'Shea, but West Ham had the result wrapped up long ago.