Hartlepool United 0 West Ham United 2
Hartlepool gave as good as they got in the first half, but two quick-fire goals before the break put the Hammers in command, the second a penalty awarded for a handball which TV replays suggested took place outside the area.
Both teams created early chances with Mark Noble curling a 25-yard shot a foot wide of the far post from the left.
Hartlepool hit back and almost took the lead on eight minutes, Scott Parker clearing off the Hammers line after a far-post header from Michael Mackay following Matty Robson's right-wing corner.
Home keeper Arran Lee-Barrett twice saved bravely at the feet of Carlton Cole as West Ham looked for a breakthrough, and Cole was booked for a late challenge on the shot-stopper minutes later.
But those two late first-half goals put the Hammers in control, Valon Behrami stroking home a low 15-yard shot on 44 minutes after Jack Collison ran at the home defence and a clearance landed at his feet.
Noble scored from the spot after Michael Nelson needlessly handled Cole's flick and that was effectively game over.
In the second half, the Hammers had the cushion they needed to allow themselves to play a relaxed game, and Hartlepool rarely troubled them.
They went close on 62 minutes when Robson crossed from the left and Gary Liddle's header brought a brilliant reflex save out of Robert Green, while at the other end the woodwork saved Pools as Lee-Barrett pushed a close-range Cole shot against the base of his right-hand post.
But two was enough for the Hammers, and they coasted home.