For much of the first half, the east Londoners did look second best, with Chelsea seemingly more risky with every attack.
With last Sunday's result, Chelsea are 10 points adrift of fourth place Spurs on the Premier League table with 57 points from 32 league games - six games before the season is over.
On a weekend that saw United return to their old habits and launch a swashbuckling comeback against a crumbling City, the former United duo added to the throwback feeling by coming on and changing the game.
Cesar Azpilicueta had fired Chelsea into a deserved first-half lead, but after they passed up a number of chances to extend the advantage, the visitors equalised through substitute Javier Hernandez. I know they are offensive players and I praised them before the game.
"Joe Hart probably needed a game like that". If we are not able to win this type of game, we are talking about what?
But, after passing up one glorious chance, Welbeck headed home the victor nine minutes from time.
Willian was inches away from doubling Chelsea's lead early in the second half after Hazard's clever back-heel gave the Brazilian space to shoot just wide.Morata has endured a hard first year with Chelsea and his fortunes weren't improving as he headed over from a corner before having another effort disallowed for offside.
But rather than discuss whether his team should have had a penalty that may have won them the match, Moyes instead insisted he would rather talk about West Ham's positive display. If it had been a really good footballing goal, which they are capable of, it would not have been as bad, but we gave away a bad goal.
He said: "Joe is having a bit of a sticky period, an inconsistent period at this moment in time, but he has got bundles of experience". The most worrying part is, it's not a surprise, we've seen this time and time again this season. Patrice Evra came in and did a very good job off the bench.
West Ham team: Hart, Ogbonna, Cresswell, Rice, Noble, Zabaleta, Kouyate, Masuaku, Arnautovic, Joao Mario, Fernandes.