After strikes from Lomana LuaLua, Sean Davis and Pedro Mendes had given the visitors an unassailable half-time lead, Bulgarian striker Svetoslav Todorov – himself a former claret and blue star – sealed the win late on to ensure that efforts from Teddy Sheringham and Yossi Benayoun proved purely consolations.
For Alan Pardew, the encounter was simply a headache he could have done without after Wednesday night's exhausting FA Cup fifth round replay win over Bolton Wanderers in extra-time.
With the quarter-final tie at Manchester City to be played on Monday night, the Hammers boss made several changes to his starting line-up, but kept his first-choice back four in place as part of his pledge to 'respect' the likes of Birmingham City and West Bromwich Albion who, deeply involved in the relegation dog-fight, will have been mortified to see their rivals secure such a resounding win.
While Pardew brought in the likes of Clive Clarke, Yaniv Katan, Christian Dailly, Bobby Zamora and Sheringham for Matthew Etherington, Benayoun, Hayden Mullins, Dean Ashton and Marlon Harewood, Redknapp named an unchanged team following Pompey's 2-1 victory over Manchester City last week - and it was clear from the word go that the visitors were more at ease with their line-up and formation.
Captain Gary O'Neil almost opened the scoring 15 minutes in, when he latched on to a cross and found himself with the goal at his mercy, but a tremendous block by goalkeeper Jimmy Walker - playing his first game back after returning from serious injury and in for Shaka Hislop, sidelined with a hamstring strain - denied what looked to be a certain goal.
Pompey didn't have to wait long for fortune to come their way though as, four minutes later, LuaLua met a Matt Taylor cross at the far post and slammed the ball into the roof of the net to open the scoring and send their travelling fans wild.
Just moments later, they were celebrating a second as LuaLua's cross from the right was reached by Davis, who burst through two challenges and lifted the ball over Walker for his first Premiership goal for the club since his January move from Tottenham Hotspur.
Prior to Saturday's game, Portsmouth had scored just two goals in their last nine Premiership matches away from home, so when they added a third just three minutes before the break, Redknapp must have thought he was dreaming.
Following his heroics against City last week, Portuguese midfielder Mendes produced another moment of sheer magic, sending a blistering 30-yard effort flying past Walker into the top corner of the net.
For the second Saturday running, Pardew's men found themselves 3-0 down at half-time and, just as he did in the 4-1 defeat against Bolton at the Reebok Stadium, it was Sheringham who offered his side a glimmer of hope in the second half.
With Harewood and Benayoun on as half-time substitutes as Pardew desperately attempted to salvage something from the game, the hosts began to fight back. After surviving a scare early in the second half when Benjani Mwaruwari prodded just wide from 12 yards out, they almost grabbed one back on 58 minutes when Harewood fired in a rasping drive that crashed off the bar.
Just ten minutes later, it was Paul Konchesky's turn to hit the woodwork, but this time his effort bounced out to Sheringham, who swept the ball home from close range for his seventh goal of the season.
However, no sooner had the Upton Park faithful began to wonder if a miraculous recovery was on the cards, Portsmouth replied with a fourth goal to restore their advantage and make sure of victory with 15 minutes remaining.
LuaLua cleverly worked his way past Konchesky, before racing to the by-line and cutting the ball back for substitute Todorov to finish off the Hammers with a simple finish.
There were late bookings for Mendes and Harewood as Portsmouth fought to protect their lead and the hosts grew more frustrated, but there was still time for one more goal, as Benayoun fired past Dean Kiely from 12 yards out after Zamora had provided a clever pass.
For Pardew's men, it is hoped that they will again recover from a Premiership hammering to produce a memorable FA Cup victory at the City of Manchester Stadium on Monday night.
For ecstatic Portsmouth, though, this was the main event, and they have given themselves a fighting chance of survival with a knock-out performance.