After taking the lead through Chris Cohen and dominating for most of the game, Kevin Keen's team saw all their hard work undone when the referee awarded an extremely harsh penalty to Everton with 15 minutes remaining, before insult was added to injury when the Toffees claimed all three points with another spot-kick in the final minute.
Following an excellent 4-1 win over Middlesbrough in their opening play-off encounter, Hammers continued where they had left off and kept Everton pinned back for much of the game.
After going close on several occasions, the hosts finally took a deserved lead in the 65th minute when Sam Wright's cross from the right was headed home by schoolboy midfielder Cohen.
With the all-important first goal under their belt, Hammers kept their shape and continued to attack in the hope of extending the lead but, with just 15 minutes remaining, all the hard work was undone in an instant.
An Everton attacker collected the ball on the left and skipped inside along the edge of the penalty area. Defender Tommy Laws read the situation and stepped across to clear the ball fairly, but no one - including the Everton players - could quite believe what followed, as the referee pointed to the penalty spot.
The Toffees converted from 12 yards and, from being in position of complete control for 75 minutes, the young Hammers were now holding on for a point and understandably losing their shape and discipline in the wake of a simply unbelievable decision.
Worse was to follow, though. With just seconds remaining, Everton pushed forward again and were awarded another spot-kick - this time a fair one - and the Hammers were left empty-handed, stunned and furious.
The result now means that the under-17s now have to beat Aston Villa away from home by at least three goals in their final group play-off match in order to progress to the next stage.
A clearly upset Kevin Keen said afterwards: "I don't really know what to say - it was a game we certainly didn't deserve to lose and I am just so disappointed for the players.
"Having dominated the game for 75 minutes, I was confident we would hold on for a deserved victory, but it was taken out of our hands by an extremely poor decision.
"I can only describe it as mystifying. The referee said that Tommy put his arm across the player, but I couldn't see that and I don't think many other people there did either. It is over now and there is nothing we can do about it, but I am just being honest and it is very upsetting.
"The players have worked hard all season and they didn't deserve that. Credit to Everton, they were very strong defensively, but the game degenerated after the penalty decision and I have no doubt that it cost us the game.
"There is just a feeling of emptiness now, but the players will keep battling and we will give it our all against Aston Villa."
Team: Reed, Laws, Blewitt, Parrington, Wright (Henry 80), Cohen, Tucker, Bunce, Ephrain, Cole, Collington. Subs unused: Shaw, Behcet, Stokes, Welsh.
Meanwhile, the under-19s rounded off their group matches with a comfortable 2-0 win over Crystal Palace.
Goals from Phil Lumsden and Daryl McMahon secured the three points after Hammers had already qualified for the semi-finals of the competition last week, and Aston Villa or Arsenal will now be the opponents in the last four.
Team: Jackson, McClenahan (Yao 64), Tattam, Norval, Fletcher, White, Lumsden, McMahon, Pearson (Reid 64), Akinsete, Singh (Allen 63). Subs unused: Andrews.