There were no nerves from the tournament newcomers as the Premier League club dominated from start to finish.
Michail Antonio showed a striker's instinct to put the Hammers ahead in the first-half by capitalising on a defensive error, and Declan Rice sealed the victory after the break with a quality solo goal.
The Hammers showed professionalism to manage their lead, and kept the ball well all night as they sailed to the top of Group H.
Story of the match
West Ham looked lively in the opening minutes, winning a couple of corners as they settled in to the game, but the early set-pieces came to nothing.
The visitors continued to control possession throughout the first 10, pressing forwards in the opposition half as they probed for an opening, the hosts remaining deep in a rigid shape.
Despite the low-block the English side found themselves facing, Moyes' men remained patient, moving the ball well across the pitch and dictating the tempo of the game.
The Croatian outfit struggled to build any momentum early on, their first effort coming from the dangerman, Mislav Orsic, however his free-kick from distance did not trouble Lukasz Fabianski.
West Ham eventually took a deserved lead just after the 20-minute mark thanks to talisman Michail Antonio, the in-form forward latching on to a loose backpass and nudging into an empty net to make it 1-0.
West Ham United (@WestHam) September 16, 2021
Theophile-Catherine's ball was underhit, and showing the same instinct that saw him take advantage of Soyuncu's mistake in the game against Leicester, Antonio beat the 'keeper to the ball before finishing with ease.
Ryan Fredericks went close to making it 2-0 moments later after he raced in behind on to Antonio's pass, but Livakovic narrowed the angle well and denied the full-back.
Despite the team's lack of European experience as a collective, the Hammers were at ease in the hostile Baltic atmosphere, their ability to control the occasion a sign of the club's progression.
Manuel Lanzini was unfortunate to pick up a harsh booking with half an hour gone, but the resulting free-kick was cleared comfortably.
The travelling 2,000 Claret and Blue faithful will have been very pleased to see their side head into the interval with a well-deserved lead, a perfect start to life on the continent.
There was one change at the break as Deni Juric replaced Theophile Catherine, the defender at fault for the goal who had a night to forget.
Zagreb made a much better start to the second half, and Petkovic fired narrowly wide just moments in, an attempt as close as any that they managed in the first 45 minutes.
West Ham had to switch on as the game's intensity went up a notch, however they were fortunate to keep eleven players on the pitch when Lanzini narrowly avoided picking up another booking with a late tackle.
Despite the hosts' improvement, Declan Rice made it 2-0 on 50 minutes with a fantastic solo goal. The 'skipper on the night showed tenacity to win possession in the opposition half, and then went on a surging run forwards before slotting under the 'keeper with his left foot.
West Ham United (@WestHam) September 16, 2021
Lanzini was replaced by Benrahma immediately after the goal as Moyes made sure to avoid any unnecessary dismissals that would allow the opposition a way back in to the game.
Ivanusec almost pulled one back for Dinamo as they looked to make a comeback, but Fabianski was relieved to see the ball trickle inches wide of his far post.
With an hour gone the home side began to pile on more pressure, the Premier League team showing the first signs of fatigue as they looked to defend their lead. Jarrod Bowen became the second West Ham substitute to be introduced as he entered for Vlasic.
In passages the Irons played some beautiful passing football, displaying the class and professionalism of an established European team as they aimed to see out the closing minutes and return to London with three points.
The triple substitution allowed the Hammers to rest key players ahead of the week, while also giving minutes to others such as captain Mark Noble, who's smile was clear for all to see as he entered the fray.
There were few chances for either side in the game's closing stages, West Ham managing the final minutes as they aimed to secure a valuable win.
Noble controlled from the middle of the park, and the instructions from the manager were clear as each player took few touches and no risks.
The Croatian supporters were relentless all night, but the cheer from the travelling Hammers was loud and clear as the final whistle ended a faultless performance for West Ham.