The Hammers, who suffered semi-final heartbreak in the continental competition just over two weeks ago, were on course to snatch sixth position from Manchester United after Michail Antonio put them ahead against Brighton.
But second-half strikes from Joel Veltman, Pascal Gross and Danny Welbeck earned the Seagulls a deserved 3-1 success to prevent Moyes’ men capitalising on United’s 1-0 loss at Crystal Palace.
The east London club must be content with a place in the Europa Conference League next term following the costly capitulation, much to the annoyance of their disgruntled manager.
“I’m just feeling there’s a missed opportunity and I’m really disappointed with our overall performance,” said Moyes.
“Second half, you could have driven a bus through us so many times and we didn’t show any real desire to make it hard for them to score against us.
“We didn’t play well and not like a West Ham team second half. Really disappointing – no resilience, no toughness, nothing about us in the second half.
“If we’re going to play like that then we’re going to be challenging in a different part of the league. I won’t have them if they’re going to play like that.”
West Ham’s trip to the Amex Stadium was their 56th fixture of a gruelling campaign, which included a two-legged loss to eventual winners Eintracht Frankfurt in the last four of the Europa League.
The visitors returned for the second half in Sussex knowing that Ralf Rangnick’s side were behind against Palace but proceeded to fall apart as Veltman swiftly cancelled out Antonio’s superb opener before Gross and Welbeck struck in the final 10 minutes.
Moyes refused to blame tiredness for his team failing to take advantage of United’s slip up at Selhurst Park.
“I wouldn’t because I would feel it would be a excuse and I won’t give the players that room,” said the Scot.
“They know I’m not happy.
“I’m not really that bothered what their mood is, but you can tell what my mood is and I’m hoping my mood is reflecting on how they’re reacting and how they feel.
“We want them to have a good holiday, but I also want them to understand that being OK in 37 games (is not enough), you’ve got to be good in them all.
“We knew (United were losing) and that’s why I was hoping I was going to see a tough, resilient, hard-to-beat team which looked as if they had something they were fighting for and I didn’t see that.”
Brighton’s comeback victory secured the highest finish in the club’s 121-year history.
The Seagulls, who could have slipped as low as 14th, climbed above Wolves into ninth spot to end in the top half of the Premier League for the first time.
Albion head coach Graham Potter, who oversaw winless runs of 11 and seven games this term, said: “I’m really proud.
“There have been moments in the season that have been challenging. I’m sure the experts out there will say, ‘should have done this, should have done that’ but generally in the Premier League it’s so difficult.
“You have to stick together, we’ve done that and to end up where we have is amazing.”