If the Clarets are to get to Wembley this season they will have to do it via the promotion play-offs after their FA Cup ambitions were blown away in a burst of second-half goals at Upton Park.
After dominating the first period with one of their best 45 minutes of the season, Eddie Howe's men looked capable of clawing back Thomas Hitzlsperger's opening goal to keep alive their hopes of an upset and a quarter-final trip to Stoke City.
But three goals in 15 minutes after the break, including a double blast from Carlton Cole, put paid to that prospect.
Carlton Cole puts the Hammers 2-0 up
Jay Rodriguez pulled one back 20 minutes from time but with Freddie Sears restoring the Hammers' four-goal deficit with virtually the last kick of the tie, the Clarets were left to digest only a second defeat in ten games.
And despite the harsh margin of defeat, Burnley have little time to lick their wounds as they re-group for Saturday's derby clash at Preston and the start of a Championship run-in which they must now try to take all the way to the Twin Towers.
West Ham's concerns are at the other end of their respective tables. But despite their travails near the foot of the Premier League, the Irons gave this their full priority with the strongest side boss Avram Grant could name sent out to avoid shock.
Yet it was the Clarets who dominated the opening exchanges as they rose superbly to the challenge of trying to reach the sixth round.
Rodriguez, who started the day by being named the Football League's young player of the month for January, was asked to lead Burnley's attack as Wade Elliott earned a re-call on the left flank.
And the striker, watched by England under-21 boss Stuart Pearce, lit up Upton Park in conjunction with Chris Eagles and Ross Wallace as the Clarets pressed for the opening goal.
Dean Marney holds off Demba Ba
A fine cross-field ball from Wallace sent Rodriguez sprinting away in the tenth minute and even though he reached the ball just before Robert Green, the angle was against the Burnley man who couldn't find the power or direction to hit the target.
His next chance took just three minutes to come, though, as Eagles and Jack Cork combined superbly to get the ball into the box where Rodriguez out-muscled New Zealand international Winston Reid and forced Green to parry his attempted lob over the bar.
Green was the busier keeper as he then had to dive to his left to keep out a stinging drive from the lively Wallace and the follow-up from Rodriguez, although the latter effort may have been ruled out by the linesman's flag.
But critically Burnley's flying start failed to bring them a goal and just as the home supporters were beginning to get restless West Ham went 1-0 up mid-way through the first half.
Summer signing Hitzlsperger had not played since his arrival from Lazio but having recovered from a thigh injury the Germany international made an immediate mark, living up to his nickname of 'The Hammer' by thumping a 30-yarder past Lee Grant.
Ross Wallace in full flight
The goal knocked some of the breath out of the Clarets but it wasn't long before they got on the front foot again, opening up the Irons' defence for Eagles to let fly with a low shot which had Green diving to his left to parry to safety.
And when Eagles, revelling on his return to the big stage, Cork and Wallace again produced some stunning football to release Tyrone Mears on the overlap, Rodriguez's near post effort was blocked off for a corner as he met the full-back's low cross in search of an equaliser just before the break.
The Clarets would have deserved to have been level but within five minutes of the start of the second half they were out of the cup as Cole struck twice within a matter of moments.
Whereas Eagles was rightly flagged for being half a yard offside at one end, Cole got the benefit of the doubt of a tight one at the other end after Mark Noble had sent him clear.
His touch as he ran into the box was bad enough to allow Grant to make a smothering save but the England international got lucky as he stuck out a leg to prod the loose ball past the covering Michael Duff and into the corner of the net.
If that was a Hammer blow for the Clarets, Cole's second goal two minutes later signalled the death knell.
Again Noble released Cole with a fine ball over the top and this time the striker showed his quality with a fine finish beyond Grant and into the roof of the net.
Jay Rodriguez feels the pain of a missed chance
Eagles tried to respond with a shot which he dragged wide but the spectre of the five goals the Clarets leaked here last season raised its head as Freddie Sears suddenly began to come alive down the Hammers' right side.
And it was when they forced a corner down that flank that the top-flight side went 4-0 up with less than an hour gone as Hitzlsperger delivered for Reid to head home.
Demba Ba then forced Grant to save as the Hammers shrugged off the cares of their relegation fight but the Clarets did at least gain some consolation when Rodriguez pulled a goal back in comical fashion.
The striker had just missed an excellent chance from Elliott's through-ball but was then in the right place on the edge of the six-yard box to nod in after Green had whacked his attempted clearance against the persistent Mears.
Food for thought
Rodriguez's eighth goal of the season capped another eventful day in his burgeoning career and substitute Chris Iwelumo almost added another when his header was somehow kept out by Reid's goalline clearance.
But the Hammers had the final say as Sears broke through in stoppage time to rattle home number five as the Londoners booked their passage to Stoke City in round six.
West Ham: Green, Jacobsen, Bridge, Reid, Tomkins, Parker, Hitzlsperger (Barrera 67), Noble, Sears, Cole (Spector 73), Ba (Piquionne 82). Substitutes: Boa Morte, da Costa, Hines, Boffin.
Burnley: Grant, Mears, Fox, Carlisle, Duff, Marney (Alexander 73), Cork, Wallace (Iwelumo 73), Eagles (Thompson 84), Rodriguez, Elliott. Subs: Edgar, Bikey, Easton, Thompson, Jensen.
Booked: Rodriguez 66, Elliott 74
Referee: Martin Atkinson.