German international Thomas Hitzlsperger, who joined West Ham in June 2010 but had been sidelined with a thigh injury since August, opened the scoring with a goal on debut 22 minutes into the fifth-round match at Upton Park.
The 28-year-old former Aston Villa man unleashed a characteristic 25-yard pile-driver to put the English Premier League club 1-0 ahead at half-time.
Carlton Cole doubled the hosts’ advantage three minutes into the second half.
The striker, capped seven times by England, had an initial shot saved by Burnley goalkeeper Lee Grant but was then able to stab the loose ball into the back of the net from close range.
Just two minutes later, Cole had a brace.
Running in behind the Burnley defence and connecting with Mark Noble’s long ball, he fired an impressive half-volley past Grant to make it 3-0.
A minute before the hour mark, debutant Hitzlsperger turned provider when he sent in a corner for New Zealand World Cup defender Winston Reid to head home from six yards out.
The visitors pulled one back in the 71st minute when West Ham goalkeeper Robert Green played a clearance straight into the head of former teammate Tyrone Mears.
The ball fell to Jay Rodriguez, who nodded it into an unguarded net.
But the strike proved no more than a consolation and Freddy Sears completed the rout in injury time with a well-taken individual effort to make it 5-1.
West Ham will now travel to fellow Premier League side Stoke in a quarter-final at the Britannia Stadium on March 12.
Their continued presence in the tournament will go some way to making up for their Carling Cup semi-final exit at the hands of Birmingham.
Nevertheless, the Hammers remain second from bottom in the Premier League and are in serious danger of joining Burnley in the Championship next season.