A double strike from Tony Stokes and one each from Hogan Ephrain and James Tomkins secured victory for Kevin Keen's team, who have scored 10 goals in their first three games of the new season.
After starting strongly, Hammers took the lead when Darren Brady swung in a free-kick from the left that was flicked into the net by Stokes after 20 minutes.
After creating two or three good chances without managing to score again before the break, Hammers eventually secured a two-goal cushion 10 minutes into the second half when schoolboy striker Ephrain cut in from the right and finished well.
Just three minutes later, it was 3-0, and this time the goal came from an even younger source. Central defender James Tomkins - who only turned 14 in March and still has two seasons of schoolboy football to play - had replaced the injured Liam White after just 15 minutes, and rounded off an impressive performance with an excellent header from Ephrain's corner.
Bristol City pulled one back late on, but Stokes restored the three-goal advantage in the final minute when he pounced on a rebound to claim his fourth goal in three matches.
Said Kevin Keen afterwards: "It was an excellent result and an excellent performance. We played some good football and the boys bounced back well after last week's defeat at Sheffield United.
"That was a tough game and a bit of an eye-opener for some of the younger lads, but they responded well today and fully deserved the win.
"We lost Liam White with a shin injury early on, but young James Tomkins came on and impressed, scoring a goal and defending very well.
"Overall, I am delighted - we were very good all over the field."
Team: Reed, Young, Norval, White (Tomkins 15), Andrews, Davidson, Singh, Cowley (Cumberworth 70), Brady, Stokes, Ephrain (Allen 80). Subs unused: Welsh, Yao.
Meanwhile, a Marce Collington goal wasn't enough to set-up a first win of the season for Tony Carr's older side against their west London rivals.
After going close on several occasions in a goalless first half, Hammers took the lead on 60 minutes when Collington cut inside from the right and curled a superb left-foot shot past the Fulham goalkeeper.
However, rather than lead to a comfortable victory for Hammers, the goal seemed to spur the visitors into life, and they equalised on a breakaway attack with 20 minutes remaining.
A draw appeared likely as the clock ran down with neither side showing any real signs of attacking quality, but Fulham snatched the win two minutes into injury time when a second successive corner was bundled home.
Said Tony Carr: "It was very disappointing to lose the game in that manner, especially as I didn't think we deserved to come away with nothing. A draw would probably have been a fair result but, saying that, we had plenty of chances to win the game.
"Greg Pearson and Marce Collington both missed very good opportunities in the first half, when we had long spells of superiority, and once we got the goal I thought we would go on to win comfortably.
"They fought back well, though, and got the breaks when it mattered. I am disappointed, but there were some good signs, and it was a big improvement on last week's performance at Sheffield United. If we can start taking our chances, I am sure the first win will come soon."
Team: Behcet, McClenahan, Tattum, Laws (Blewitt 80), Lumsden, Tucker, Wright, Bunce, Cohen, Pearson (Lee 75), Collington (Carrick 75). Sub unused: Parrington.