Goals in the opening 10 minutes of each half, firstly from Jarrod Bowen and then a header from Michail Antonio, give West Ham a routine route into round five.
It will get much tougher in the next round for the Hammers though. Manchester United away.
James Collins crosses from the right for Derby, but Haydon Roberts, under pressure, can't get a clear header at goal.
Three minutes of injury time, but it's been all so routine since that second West Ham goal.
There has been a lot of endeavour, heart and desire from Derby.