Tony Carr's team made a bright start against last season's FA Youth Cup finalists, and took a deserved lead after just four minutes, when Daryl McMahon found the net with an excellent shot from the edge of the area.
The Hammers then dominated for the rest of the first half and should have added to their tally as Freddy Eastwood and Elliott Ward both had efforts cleared off the line, while James Allen and Phil Lumsden also came close.
After denying their hosts a vital second goal, Everton grew in confidence after the break and tested Glenn Jackson on a couple of occasions.
However, Hammers were always in control and, despite not adding to their scoreline, held out for a deserved victory.
Said Tony: "It was a good victory but we really should have won by more. At 1-0 it was a bit edgy and we needed that second goal to settle us down.
"I don't think I've ever seen a game where we've had so many shots - there must have been about 30 in total - and we really should have made more than one of them count.
"The important thing is that we got the three points, though, and it's pleasing to know that we can grind out a result like that when things aren't really going our way in front of goal.
"Everton fielded the majority of the team that reached the final of the FA Youth Cup last season after beating us along the way, so it was a very satisfying win."
Team: Jackson, Khan (McClenahan 75), Tattam, Ferdinand, Ward, Lumsden (Pearson 75), Allen, McMahon, Mehmet, Eastwood (Morris 80), Cole. Subs unused: Sealey, Fletcher.
Meanwhile, Kevin Keen's under-17s suffered their second successive defeat against Merseyside opponents, following last week's opening day loss at Liverpool.
After an evenly-matched opening 45 minutes, the Hammers began the second half in impressive fashion, but couldn't find the all-important breakthrough and were made to pay as Everton scored from a corner with 20 minutes remaining.
The youngsters battled on in search of an equaliser, but their efforts were to no avail, as the visitors broke away in the final minute and won a penalty that was converted to put the result beyond doubt.
Said Kevin afterwards: "Last week I was disappointed with the result at Liverpool, this week I am very, very disappointed. Once again, I couldn't fault the effort of the players and some of our football was very good in the midfield area.
"However, we just couldn't seem to find that extra bit of quality needed in the final third of the field and failed to really trouble their goalkeeper.
"That said, there are some positive signs to come out of our performance and it is important that the lads don't allow their heads to drop. They are all working very hard and I'm sure results will improve once they get more used to playing together as a team."
Team: Behcet, Wright, Blewitt, Laws, Parrington, Tucker (Henry 80), Yao (Stokes 70), Bunce, Lee, Collington, Cohen. Subs unused: Shaw, Wayman, Singh.