Goals either side of half time from Jarrod Bowen and Michail Antonio helped the Hammers cruise past their League One counterparts to set up a fifth-round tie against David Moyes's former side Manchester United at Old Trafford towards the end of February.
Derby headed into the fourth-round contest on home soil in fine form after putting together a 14-game unbeaten run in all competitions, while West Ham had only won two of their last seven games before their trip to the East Midlands.
However, it was the Hammers who made a bright start to proceedings and they opened the scoring in the 10th minute when Tomas Soucek ran onto Antonio's looping through-ball before flicking a header towards Bowen to poke a first-time shot past Joe Wildsmith from inside the six-yard box.
Derby responded well and looked particularly lively down the right flank with Nathaniel Mendez-Laing, but the Rams were unable to test Alphonse Areola in the West Ham goal and failed to register a single shot on target in the first 45.
Paul Warne will have hoped his team could find their creative spark and stay in the tie heading into the closing stages of the second half, but they were ultimately facing an uphill battle just five minutes after the break after Antonio doubled West Ham's lead.
After Derby were caught short in numbers at the back, the ball found its way to Bowen on the right channel, and the Englishman's cross took a slight deflection off Craig Forsyth's outstretched leg before floating towards Antonio to nod a simple header in at the far post.
There was a potential penalty claim for Derby on the hour mark when Ben Johnson clumsily barged into Curtis Davies during an aerial challenge just inside the box, but with no VAR the decision was waved away by the on-field officials.
Everton loanee Lewis Dobbin, one of four Derby changes made on the 65-minute mark, whipped an inviting delivery towards the back post shortly after coming off the bench, but all three Rams players inside the area were unable to meet the cross.
Three minutes later, Wildsmith was called into action to deny Bowen with a strong save from around eight yards out, before Derby's final substitution saw Jake Rooney - the cousin of former Rams player and manager Wayne Rooney - brought on to replace Connor Hourihane.
West Ham dominated the closing stages against a youthful Derby outfit and saw out the match to claim their third win in four games.
Moyes and co will now turn their attention back to the Premier League with a challenging contest away against high-flying Newcastle United on Saturday, while Derby will play host to Morecambe in League One.