Mohammed Salah got the Reds' season up and running with a goal early on, before a Sadio Mane brace and a late Daniel Sturridge strike saw Jurgen Klopp's men take all three points, convincingly.
Klopp's men quick out the blocks
It has been a summer full of excitement at the London Stadium, with a change in management as well as nine new first-team signings, although the Irons were brought right back down to earth from the very first minute of the season as the Redmen fired out of the blocks from the off, hardly giving Pellegrini's men a kick.
New signing Naby Keita looked very impressive for the home side from the off, winning everything in midfield as well as acting as the catalyst for Liverpool - releasing the ball to the front-line time and time again.
Following a dominant first 19 minutes, which saw all three of Liverpool's front men coming close, the deadlock was broken by last seasons PFA Player of the Year, Salah - a tap-in for the Egyptian following a menacing run by Keita, which was followed by a great ball in from Andy Robertson.
Following the goal, there was minimal reaction from the Hammers, who simply couldn't get a foot on the ball, resulting in only one half-chance in the first-half, with the ball dropping to Marko Arnautovic who dragged his effort wide.
Man of the Match Mane?
Either side of half-time, Sadio Mane added to Liverpool's lead, putting the game beyond any doubt.
The Senegalese internationals first saw yet another cross in from the impressive Robertson, before the ball was pulled back by the stretching James Milner for a tap-in for Anfield's new number 10.
The second 45 saw more of the same, Liverpool quick out of the blocks and scoring through Mane again. A question of offside as the away side tried to hold a high line, but it was again too easy for Klopp's men, as Roberto Firmino slipped the ball through to Mane for another routine finish.
Despite his goals, Liverpool's left-winger was impressive going forward all afternoon and had to be a strong contender for Man of the Match.
Sturridge caps off a brilliant performance
3-0 up, and Liverpool just wanting more of the same, attack, attack, attack. Sturridge rounded off the scoring with his first touch, getting on the end of Milner's corner to complete a terrific afternoon for the home side, and send them top of the Premier League.
The East-Londoners shouldn't look at this game as one that will define their season, coming up against a Liverpool outfit who couldn't have looked any more in the mood, however, at times Pellegrini's men's defending was questionable and is maybe something to focus on in training in the coming week.
New record signing Felipe Anderson saw very little of the ball, and will undoubtedly be able to showcase his talent more effectively in coming weeks as the fixtures become easier than this opener.
Next up for Liverpool is a tricky trip to Selhurst Park to face Crystal Palace, as the Hammers look to get their first points on the board at home to Bournemouth.