A double strike from Billy Mehmet and one from Mitchell Cole gave Tony Carr's team victory in the Potteries. The Hammers took the lead in the 10th minute, when a splendid through-ball from Anton Ferdinand released Mehmet, who finished well.
Stoke equalised against the run of play after 25 minutes, but the visitors should have restored their lead before half-time, with Cole and Phil Lumsden both missing good opportunities.
However, Hammers dominated again in the second half and added a second when Terry Khan's pass set Mehmet through on goal again, and the young striker doubled his tally.
With 15 minutes remaining, Cole settled the match when he cut in from the left and fired a shot into the far corner of the net.
Said Tony Carr: "It was a good performance from the boys against a hard-working Stoke team. The goal we conceded was disappointing and we wasted some excellent chances, but the players kept their concentration and the second half display was very pleasing.
"It's nice to start the season with a good win and that will hopefully set us up for a successful campaign."
Under-19s Team: Jackson, Khan (Fletcher 81), Tattam, Ward, Ferdinand, Lumsden, J.Allen, McMahon, Mehmet, Eastwood (Pearson 76), Cole (Akinsete 76).
Meanwhile, Chris Cohen's goal was not enough for Kevin Keen's under-17 side, who put up a spirited fight against an excellent Liverpool side up in Merseyside.
Despite being the better side in the opening 20 minutes, Hammers went a goal down midway through the first half and then suffered a crucial blow on the stroke of half-time when a free-kick from 25 yards out was deflected off the wall and in to the top corner to double Liverpool's lead.
Our youngsters fought back well after the interval and reduced the defecit through Cohen on 65 minutes. After that, however, the game became rather niggly and scrappy and Liverpool added a third in the closing minutes.
Said Keen: "It was a disappointing result, as we desperately wanted to take something from the game, but I was very pleased with the effort that the lads put in.
"There second goal right on half-time was a killer blow but I thought that we battled back well and were unlucky not to get something from the game.
"It was always going to be a tough game for us but there were plenty of positives to take from it and we can go into next week's game against Everton with confidence."
Under-17s Team: Behcet, Wright, Laws, Parrington, Blewitt, Shaw (O.Allen 83), Tucker, Bunce, Cohen, Lee, Collington (Yao 60).