Bolton's victory was sealed before half-time with three strikes in an impressive opening 45 minutes.
Gary Speed and Stelios celebrate Bolton's opener
Wanderers were ahead on 12 minutes when Stelios slotted into an open net after Ricardo Vaz Te's header hit the post.
They went two up when Stelios deflected in Kevin Nolan's drive before Gary Speed's acrobatic flick made it three.
Teddy Sheringham pulled back a consolation goal 12 minutes from time but substitute Henrik Pedersen completed the rout in style late on.
With both sides due to meet each other again on Wednesday in the fifth-round FA Cup replay, both teams were keen to secure a psychological advantage at the Reebok Stadium.
And the home side were ahead after just 12 minutes when Ricardo Gardner's cross from left-back found Vaz Te who beat Shaka Hislop in the air and headed against the post.
The rebound fell kindly inside the area for Stelios who slotted home into the open net.
Buoyed by their early opener, Bolton went in search for their second and completely dominated the first period.
West Ham's first and only opportunity of the half came when Sheringham and Bobby Zamora linked up well to provide Shaun Newton with a half chance which Joey O'Brien scrambled clear.
And it was Bolton who found the net again after 33 minutes when Stelios stuck out his leg and diverted home Nolan's long-range effort.
Referee Mike Dean had to receive treatment after Anton Ferdinand clattered into the official.
A fierce first-time shot from 25 yards almost gave Stelios his hat-trick 10 minutes before the break and Hislop did well to twice deny Jay-Jay Okocha from outside the area.
And Bolton's superiority was rewarded with a third on the stroke of half-time when Speed acrobatically turned in Gardner's header to put Bolton firmly in control.
West Ham introduced Yossi Benayoun and Christian Dailly at the interval in an attempt to claw back the huge deficit.
But Wanderers continued to threaten in the second half with Stelios and Nolan both going close.
With the game turning scrappy, the Hammers were unable to find any momentum and Bolton looked to see out the clock.
The visitors finally had something to cheer when Benayoun's shot was parried by Jaaskelainen leaving Sheringham to tap in from close range 12 minutes from time.
But Bolton had the final say when Pedersen's delightful stab from outside the box gave Hislop no chance.
Bolton manager Sam Allardyce:
"It was a terrific performance from start to finish.
"The first 45 minutes were probably the best we've produced all season in terms of flair, ability, passing and movement and ultimately the finishing.
"Even at 3-0 I felt luck was on their side, the result didn't flatter us by any stretch of the imagination."
West Ham manager Alan Pardew:
"I made some tactical changes, thinking they would make us stronger defensively but it did not work out like that.
"I will not lay any blame at any of the player's door. I have to take full responsibility for the defeat.
"We did not look sound out there today and Bolton did play well. We just had one of those games."
Bolton: Jaaskelainen, O'Brien (Hunt 67), Ben Haim, Jaidi, Gardner, Nolan, Okocha (Nakata 84), Speed, Giannakopoulos (Pedersen 75), Davies, Vaz Te.
Subs Not Used: Walker, Borgetti.
Booked: Vaz Te.
Goals: Giannakopoulos 12, 33, Speed 45, Pedersen 81.
West Ham: Hislop, Scaloni, Gabbidon, Ward (Dailly 45), Konchesky, Newton (Benayoun 45), Ferdinand, Mullins, Etherington, Sheringham, Zamora.
Subs Not Used: Bywater, Ashton, Harewood.
Booked: Scaloni, Sheringham, Konchesky, Dailly, Ferdinand.
Goals: Sheringham 79.
Ref: M Dean (Wirral).