The Hammers are now three points clear of the drop zone after their first victory since the restart thanks to Andriy Yarmolenko's dramatic last-gasp winner at the London Stadium.
West Ham head to in-form Newcastle on Sunday, but after that have a favourable looking run-in including matches against fellow relegation battlers Norwich, Watford and Aston Villa.
Hammers boss Moyes said: "Well, one win is not enough. It's only one win, we have to win more if we are to remain a Premier League club, but picking up wins was going to be hard.
"You all understood the difficulty of our first three games against Wolves, Spurs and Chelsea.
"Then you saw how Chelsea played against Manchester City in midweek and they were outstanding – but this was an outstanding performance and result by our players."
Defeat meant Chelsea missed the chance to leapfrog Leicester into third.
Manager Frank Lampard said: "We have six games ahead of us now and this wasn't a huge surprise.
"If you play in the Premier League and make mistakes, if you boss possession but don't do quite enough with it at the top of the pitch, then it can happen."