After losing only one of their previous 13 FA Cup matches played on a Monday, West Ham should have been ahead after nine minutes, only for Jarrod Bowen to take on the shot himself instead of playing in strike-partner Michail Antonio, who had found himself in acres of space in the area.
Bowen redeemed himself less than 60 seconds later, as the England international culminated an exquisite Hammers team move with a clinical strike from close-range to give his side the lead.
In pursuit of their fourth successive success in England's most historic club knockout tournament for the first time since 1976, the Rams’ heads did not drop as Conor Hourihane fired an ambitious effort wide of the post.
Meeting in their first head-to-head encounter since Derby County's 2-1 victory during the 2010/11 Championship campaign, West Ham doubled their advantage in the opening stages of the second half as Antonio headed in Bowen’s pin-point cross at the back post.
Needing a change in fortunes to salvage a replay at the London Stadium, Rams manager Paul Warne introduced an exuberance of youth off the substitutes’ bench onto the Pride Park turf shortly after the hour-mark.
Despite introducing fresh faces, the visitors came closest to adding to the scoreline as Rams goalkeeper Joe Wildsmith denied the goal-hungry Bowen with a sublime stop with the last significant save of the contest.
West Ham will now grace the fifth round of the FA Cup for the third successive season as Moyes's side now prepare for the excursion to face Erik ten Hag's side at Old Trafford towards the end of the next month.
Meanwhile, Derby County will now turn their attention to attempting to reduce the gap between themselves and Sheffield Wednesday, who occupy the final League One automatic promotion position.
Flashscore Man of the Match: Michail Antonio (West Ham United)
