In a lively and open encounter at Upton Park, first half goals from Elliott Ward and Phil Lumsden were enough to see the Hammers through, but it wasn't a smooth passage for Tony Carr's team.
Norwich will consider themselves unlucky not to be in the last eight after enjoying the better chances in front of goal over the course of 90 minutes, but that was the crucial difference between the two sides - Hammers took their chances, Norwich didn't.
Clear-cut opportunities were at a premium for West Ham's strike-pairing of Danny Morris and Greg Pearson, but it was the hosts who took the lead in the 21st minute, when Ward fired home from the penalty spot after Pearson had been fouled in the area as he cut in from the left.
Norwich responded well and got in behind the Hammers' defence four times in the space of 15 minutes, but were denied by goalkeeper George Sealey - son of our late former goalkeeping coach Les - who was in fine form all evening.
Hammers finished the half strongly, and claimed a vital second goal just two minutes before the break, when Ward's free-kick from his own half sailed all the way over a static Norwich defence and released Lumsden, who composed himself in acres of space before firing confidently past keeper Osborne.
The hosts might have made it three before the half-time whistle blew, when winger Mitchell Cole found himself in a good position on the edge of the area, but fired a stinging shot into the side-netting.
The second half took on a similar pattern to the first, with both sides finding it hard to keep the ball and place any sustained pressure on the opposition defence.
Pearson went close with a shot from 20 yards out that was saved well by Osborne in the 55th minute, while Norwich's greatest threat came from their skilful number seven, Ian Henderson, and blond striker Ryan Jarvis, who showed some outstanding touches and was probably the best performer on the night.
Indeed, it was Jarvis who pulled his side back into it on 70 minutes, collecting the ball some 25 yards out before curling a beautiful shot into the top right hand corner.
That appeared to set up a thrilling last 20 minutes, but central defenders Ward and Anton Ferdinand stood firm to keep the Canaries out, and it was Hammers who enjoyed the better chances in the closing minutes.
Substitute Trent McClenahan burst clean through in the 80th minute but fired his shot straight at Osborne from the edge of the area, while Morris also beat the offside trap in added on time, but was again denied by the Norwich keeper as he raced into the area.
Despite there being few stoppages, an amazing six minutes of injury time was played, but Hammers held on for the victory and will now travel to Manchester City or Millwall in the quarter-finals.
Team: Sealey, Laws, Tattum, Ferdinand, Ward, Lumsden, Collington (McClenahan 73), Carrick, Morris, Pearson, Cole (Bunce 73). Subs unused: Behcet, Blewitt, Wright.