West Ham United 3 Brighton & Hove Albion 0
After a blank opening period, teenage terrier Noble cured his side's New Year hangover with his first-ever goal for his beloved Hammers, before Cole and Mullins finally sunk battling Brighton.
Following their right Royal work-shy 6-0 walloping at Reading on New Year's Day, Alan Curbishley had made one change per goal as Noble, Cole, goalkeeper Roy Carroll, George McCartney, Carlos Tevez and new-signing Luis Boa Morte comprised the half-dozen replacements.
The Hammers, who were knocked out of the Carling Cup by Chesterfield in October, may have been sitting 42 places above the visitors but if ever there was a better time for the Seagulls to leave West Ham - with just five wins all season - in yet more mess, then this was surely it.
But following their resounding FA Cup victories over non-leaguers Northwich Victoria (8-0) and Stafford Rangers (3-0), last year's beaten finalists were always going to present a stiffer test for the south coast side.
As early as second minute, though, Alexandre Frutos defiantly deposited an angled 20-yarder onto the roof of Carroll's net and, in a feisty opening half-hour, both Kerry Mayo and Adam El-Abd were booked for fouls on Yossi Benayoun and Boa Morte.
Dean Wilkins had made four changes following Monday's narrow defeat at Bournemouth as Mayo, Frutos, Joel Lynch and Joe O'Cearuill returned to a Brighton side that restricted the confidence-sapped Hammers to just a Tevez shot, a Cole header and a Boa Morte curler during a rain-swept, goalless first half.
And with Anton Ferdinand retiring with a groin injury on the stroke of half-time, things were looking pretty bleak for Curbishley, whose team had clearly not shaken off the effects of their disastrously dodgy start to 2007.
Within five minutes of the restart, however, Noble lifted all the gloom and doom when he rifled home from 12 yards after Tevez cleverly nodded Guy Butters headed clearance into the youngster's path after Boa Morte crossed from the left.
Brighton almost conjured up an instant response, when substitute Gary Hart sent a right-wing cross skidding into the six-yard box but, diving headlong with Christian Dailly, Alex Revell agonisingly failed to make contact.
That was to prove their last chance to save the tie, for when Boa Morte whipped over another tantalising centre on 58 minutes, Cole sent a 12-yard, side-footed volley ripping into the Brighton net as the static visitors' defence appealed in vain for an offside flag.
With the pressure off at last, and the Hammers ball all but in the fourth round draw, former Seagulls' striker Bobby Zamora stepped from the bench to a rapturous reception from the away supporters, while man-of-the-match Tevez received just as much appreciation from the home crowd as he teased and tormented his way to the final whistle.
And well into stoppage time, Mullins added an over-inflated edge to the final score-line when he rose highest in the six-yard box to glance home substitute Shaun Newton's left-wing cross.