But Hartlepool, who defeated Stoke in the last round, refused to lie down and were the better side for most of the second half.
Hammers boss Gianfranco Zola respected the threat posed by their opponents by selecting nine of the team that defeated Fulham 3-1 last weekend, with only Lucas Neill and Matthew Upson dropping out.
And Zola's team settled quickly, showing a crispness and composure in possession.
Hartlepool defender Nelson almost gifted the visitors an early goal with a defensive header that drifted towards his own goal but a combination of keeper Lee-Barrett and the crossbar spared his blushes.
Noble went close with a strike across goal while Herita Ilunga enjoyed plenty of space down the left flank.
But the north-east club forced their way into the contest, catching the eye with their precise approach play and desire to run at their opponents.
A Michael Mackay header caused alarm in the West Ham defence, with acting skipper Scott Parker scrambling across goal to clear.
The crowd really got behind their side but the Hammers, having survived a period of Hartlepool pressure, took control of the tie.
Lee-Barrett twice saved bravely at the feet of Cole and then felt the striker's studs in his chest as he once again tried to beat him to the ball. Several Hartlepool players reacted angrily, while Ilunga unnecessarily waded into the developing melee. Both Cole and Ilunga were booked by referee Lee Mason.
Behrami opened the scoring after Nelson had half cleared from Jack Collison shortly before the break.
And the decisive second came deep in first-half injury time as Noble struck from the penalty spot following Nelson's handball.
Despite the double blow, Hartlepool took the game to their Premier League opponents after the break and Green had to make a superb reaction save to prevent Liddle from scoring with a header, while Ritchie Jones shot narrowly over.
West Ham's superior quality only occasionally showed itself, but a through ball from David Di Michele set Cole clear, though Lee-Barrett half-stopped his strike, which then rebounded to safety off the post.
Hartlepool boss Chris Turner:
"There were a number of key decisions and they did not go for us.
"They were given a penalty but the ball was not handled in the box and that second goal made it a lot easier for West Ham.
"It made it a lot harder for our boys going out in the second half because it was a mountain to climb psychologically.
"But in both halves of the game, the first 25 minutes of each half, we took the game to West Ham, caused them problems and put in a very creditable performance."
West Ham manager Gianfranco Zola:
"It was the result we were looking for and we are pleased.
"It wasn't, especially in the second half, a brilliant performance, but maybe we thought we wanted to keep it that way and not risk too much.
"But it was a great result. That's what we wanted and it was very professional."
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