It would not have been the start Gianfranco Zola would have wanted, having gone without a win in five games, but substitute Collison rattled them into action.
Matthew Upson was stretchered off after just 17 minutes following an awkward fall jumping for a header, but Collison showed great energy and invention in midfield, with Lucas Neill pushed into central defence.
Bellamy was causing the Everton defenders all sorts of problems on both wings and after a fine run on the left pulled the ball back for Freddie Sears to line up his shot from the edge of the box.
Zola finds Hammers 'frustrating'
The 18-year-old reacted brilliantly, only to see his rasping shot tipped on to the bar by Tim Howard.
As half-time approached it was one-way traffic, and Sears wasted another good chance from six yards after good work on the left by Luis Boa Morte.
It must have been a lively half-time team talk by Moyes because his players looked transformed and were at last stringing passes together.
After a dangerous Mikel Arteta corner, Jack Rodwell rose above the static defence but could only head wide of the right post.
The Hammers then grabbed a thoroughly deserved goal and it was a goal worth waiting for.
Bellamy, once again, burst down the left with intent and threaded the ball through to the impressive Parker who flicked the ball sweetly into the path of 20-year-old Collison.
The homegrown academy player showed maturity beyond his years as he curled the ball into the top-right corner to give Howard no chance.
So with seven minutes left and Moyes prowling on the touchline, the recovery began. Saha crossed from the right and, with the defenders nowhere to be seen, Lescott headed easily down into the net.
Everton fightback pleases Moyes
West Ham looked shell-shocked, and two minutes later Anichebe crossed for Saha, whose deflected shot edged past Robert Green.
Another two minutes and it was three. Julian Faubert carelessly cleared the ball into the path of Saha, who looked up and calmly curled the ball inside the right post.
West Ham may well be advised to leave it a few days before chasing up that invitation for another visit from Barack Obama.
Everton boss David Moyes on Louis Saha:
"He is a top talent. He has not missed a day's training since we signed him.
"If he can continue to get goals then it will help him and us. He is on a two-year contract and signing him was a bit of a 'no-brainer' for us.
"The only question was his fitness. I think it was a little bit of a gamble as we've tried not to sign too many players with an injury record.
West Ham boss Gianfranco Zola:
"It's very difficult to accept. It is not the first time we've spoilt a good job in the last 10 minutes of a game.
"It is something that concerns me very much, obviously. The only thing I can say right now is that we need to find a solution.
"Maybe I have to give something more because the Premier League is punishing. You cannot afford to drop your attention for a minute."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.