Despite taking a first-minute lead through David Bentley, Lucas Neill's effort was a mere consolation as Rovers simply capitulated under the weight of Teddy Sheringham's penalty, Zurab Khizanishvili's own goal and strikes from Matthew Etherington and Bobby Zamora.
Following Monday night's 2-1 win over Fulham, Alan Pardew made three changes as keeper Shaka Hislop replaced the injured Roy Carroll, one-time £5.3m Rovers' signing Christian Dailly came in for the Prague-bound Tomas Repka and reported Blackburn and Tottenham Hotspur target Sheringham got the nod over Marlon Harewood up front.
Mark Hughes made two switches following the ill-tempered midweek Carling Cup semi-final defeat at Manchester United as Andy Todd and Aaron Mokoena replaced Ryan Nelsen and Tugay.
And on a freezing East End afternoon, Rovers quickly caught West Ham cold when Shefki Kuqi ghosted, unchallenged, down the right wing and crossed into the six-yard box where Bentley back-heeled the ball over the line, to give his side the lead with just 26 seconds on the clock.
Although Hayden Mullins tested Brad Friedel with a low 20-yarder, it was proving to be a frustrating opening for the Hammers who saw Dailly collect his sixth booking of the season for scything through Steven Reid on the quarter-hour mark.
Indeed, there could have been even worse trouble for Pardew's men, for from the subsequent free-kick, Andy Todd's point-blank header had to be acrobatically saved by the alert Hislop.
The Eastenders had barely threatened, but suddenly the game was turned upside down inside four frenetic minutes, on the half-hour mark.
First, Neill crudely assaulted Etherington on the left flank and when Paul Konchesky floated over the resultant free-kick, Yossi Benayoun's flick was handled by Khizanishvili as he held off Zamora.
Referee Martin Atkinson instantly pointed to the spot and the fit-again Sheringham showed just why, at 39, he is still such a sought-after commodity as he floated a cheeky chip over the diving Friedel.
And as Blackburn reeled from the shock of losing their stranglehold on the tie, Zamora then unselfishly invited the supporting Etherington to drill a low ten-yarder beyond the exposed Rovers' keeper to give the Hammers an unlikely interval lead.
In first-half stoppage time, Hislop brilliantly clawed out Mokoena's near-post header and just after the restart, the Hammers' keeper denied him again when he diverted the midfielder's scorching 20-yard shot over the bar.
But just as Rovers looked to be clawing their way back into the match, on the hour, Sheringham cleverly picked out the overlapping Konchesky, whose low centre into the goalmouth was turned into his own net by the luckless Khizanishvili under pressure from Benayoun.
That was the cue for Hughes to make a double substitution as Brett Emerton and Jemal Johnson entered the fray.
It was a switch that paid instant dividends for, within just five minutes, Neill curled an exquisite 20-yarder under Hislop's left-hand angle.
That still gave Rovers hope of forcing an Ewood Park replay, but as the flickering Upton Park floodlights started to fade, with a quarter-of-an-hour remaining, Zamora then put the lights out on Rovers' FA Cup aspirations when he tapped home the rebound after Friedel could only parry substitute Harewood's stinging long-ranger.