Winston Reid also netted his first goal for Avram Grant's side
Carlton Cole hit a two minute brace – the first rather dubious, with strong calls for offside and a handball before the England man bundled the ball home - at the start of the second half, which quickly took the wind out of the Burnley sails, who looked as if they could get back in to the game until that point.
Winston Reid made it four for the Hammers with a headed goal from a Hitzlsperger corner, before Jay Rodriguez pulled one back for the Clarets, but it was too little too late as Burnley crashed out of the cup.
Freddie Sears made it 5-1 to the Hammers, capping off a night that Burnley and Eddie Howe would rather forget.
But Burnley had started the match brightly, with Chris Eagles in particular looking lively for the Championship side.
Nonetheless, it wasn't long before the hosts started to up the pace. Cole embarked on a storming run down the right wing just after the five minute mark, before putting in a cross which found Germany international Hitzlsperger in the Burnley penalty, who scuffed his attempt which was easily collected by Lee Grant.
Scott Parker has the next attempt on goal, finding himself in possession a yard outside of the Claret’s 18-yard box. Under pressure from Eagles, the Hammers captain missed his mark by some distance.
Rodriguez had Burnley’s first shot on goal, taking a looping shot close in to the West Ham goal after some poor defending from Reid – who failed to clear his lines – and forced Rob Green to tip the ball over the cross bar.
Ross Wallace managed to turn Wayne Bridge on the edge of West Ham’s area and thundered a low shot which Green parried well. Mere seconds later, Green’s reaction saved the Hammers again, scrambling the follow up away – though the assistant referee’s had all ready flagged for offside.
The Clarets then enjoyed a spell of dominance until a wonder strike from Hitzlsperger saw the Hammers go 1-0 up.
West Ham were content to sit back and play counter-attacking football, using the pace of their front men to catch Burnley on the break. Scott Parker slid a simple pass through to ‘Der Hammer’, who hit the ball from 20 yards out – beating Lee Grant in the Burnley goal who got a hand to it, but could do little to stop the net from rippling.
Eagles then had a chance to level the game for the Clarets with just over ten minutes remaining in the first half with a strike from distance of his own. He was put in by Daniel Fox on the outskirts of the West Ham area and drove a low shot directly at Green, who failed to hold the ball, but luckily for the Hammers there were no Burnley players in the box for the follow up.
Wallace put a ball into the channel on the right wing for Burnley, Tyrone Mears got on the end of it and managed to put a dangerous drilled ball across the home team’s six-yard box which was hastily cleared, seemingly by both of the West Ham centre halves.
The first half ended with a higher tempo West Ham. Sears in possession was held up by Fox down Burnley’s left flank, and passed to Lars Jacobsen before finding himself some space just inside of the Claret’s area, but screwed a low shot across the opposition goal which passed the wrong side of the post and out for a goal kick.
Cole put the Hammers 2-0 up just two minutes in to the second half. Receiving the ball in what appeared to be an offside position, he seemed to lose control with his second touch. Grant got to the ball, but couldn’t hold onto it, and despite suspicions of handball by the striker, Cole managed to poke the ball into the net with his left foot.
And it was two in two minutes for Cole as he received a long ball from Noble. There was nothing dubious about this goal however, as Cole controlled the ball well and calmly side footed a lobbed shot over Grant in the Burnley goal making it 3-0 to West Ham.
Eagles had Burnley’s first decent chance of the half, creating space for himself a yard out of the Hammers’ box but dragging the shot wide to the left of Green’s goal.
The game was quickly turning into a rout and just before the 60 minute mark as Reid nodded home through the arms of Grant in the Burnley goal from a Hitzlsperger corner.
It could have been five just two minutes later as Sears, who had drifted in to a more central position than he had been previously playing in to set up Demba Ba. The January signing took a touch in the Clarets' area before blasting a shot along the ground which Grant managed to get his finger tips to.
Hitzlsperger's debut was cut short on 66 minutes, with his manager clearly wanting to rest the player - having just recovered from injury - and the German was met by a standing ovation from the Upton Park faithful.
Confusion between Bridge and Green saw Burnley awarded a consolation in the 70th minute. Bridge failed to clear a ball rolling parallel to the dead ball line and Green’s clearance rebounded off of Tyrone Mears at point blank range, leaving the ball to be nodded in by Rodriguez from two yards out.
Grant was called upon again with ten minutes left when Demba Ba, breaking into the Burnley box, took a shot on the half volley which the Claret’s keeper did well to turn over the bar.
It was made 5-1 by Sears in the 92nd minute after picking up the ball from the feet of teammate Spector, and making a run in to the Burnley box and blasting the ball to the left of Grant and in to the Burnley net, capping off an embarrassing evening for the Championship side.
West Ham will now be taking a trip to the Britannia Stadium to take on Stoke in the next round.