The experienced duo, accomplished Premier League players over recent seasons, headed home in the first half after Forest captain Luke Chambers had put through his own goal.
The hosts hit back when substitute Joel Lynch fed Robbie Findley to reduce the deficit midway through second half, but Winston Reid's goal left manager Steve McClaren in no doubt about the huge task facing him at the City Ground when he tapped in a fourth for the Irons with 13 minutes remaining.
With both sides heavily backed for promotion in pre-season, West Ham are already demonstrating their intent to secure a return to the top flight, while Forest continue to flounder in front of a frustrated home faithful.
Despite a run of only two losses in their opening seven competitive matches, McClaren's men remain winless by the River Trent so far.
A third win in a row on the road represents a real achievement for United and their manager Sam Allardyce, especially in light of the fact they had taken just 12 points from their last 39 road trips.
Influential midfielder George Boateng returned to the home line-up, with Allardyce recalling his big names, Mark Noble and Cole, after a shock exit from the Carling Cup at the hands of Aldershot.
Yet there was no place for England midfielder Scott Parker, expected to leave the club imminently, or for new attacking option Sam Baldock who completed his transfer from MK Dons on Friday.
West Ham bared their teeth after only three minutes, Julien Faubert snatching at a close-range opportunity after Cole had robbed Chambers of possession.
Findley got his head to Lewis McGugan's outswinging corner in the 12th minute but nodded wide of the upright, while the Hammers surged forward from the restart, Cole responding with a glancing header that went wide following Joey O'Brien's early cross.
Yet the Hammers were ahead in the 21st minute, Matt Taylor whipping the ball into the six-yard box, only for Chambers to fumble it into his own net.
Nolan used his head to double the advantage three minutes later, James Tomkins nodding the ball into a throng of visiting players for the former Newcastle midfielder to bag his second goal this term.
Forest had two opportunities to reduce the deficit almost immediately. Ishmael Miller sprang clear on the right wing but stubbed his shot into the side netting, before Rob Green denied McGugan following Chris Gunter's weighted cross.
But Allardyce's men heaped more misery on McClaren's team in the 32nd minute, Cole nodding home Reid's pacey cross.
Nolan spurned a golden opportunity to make it 4-0 in the 42nd minute, inexplicably slamming the ball wide of goal after Taylor cleverly dummied Cole's low cross.
West Ham seemed content to sit deep and keep the frustrated Reds at arm's length as the second half began, prompting McClaren to beef up his creative options by introducing summer signings Andy Reid and Matt Derbyshire in the 63rd minute.
The change paid dividends seven minutes later, Reid and Lynch combining to slot a ball across the box to Findley who eagerly slotted home.
But the comeback was dead and buried in the 77th minute when Reid latched onto Taylor's pinpoint cross to slip the ball beyond Lee Camp.
The home fans might have expected more when Reid stung Green's palms from close range on 87 minutes, but despite five minutes of added time, the Reds' fate was sealed.