Moyes made the admission in today's match programme notes for the early kickoff against Manchester United.
The Scot wrote: "Welcome back to Goodison Park for today’s early Premier League kick-off.
"Including today, we now have only six games left at this wonderful old ground and, just like last week, we face another big opponent.
"Last Wednesday we found ourselves in the final Merseyside derby of the Goodison era and I have to say it was an incredibly memorable night.
"The only thing which might have made it better was us getting three points from the match.
"The players put in huge levels of effort to give us every chance of winning the game and all credit to them for keeping it going right to the very end. Our performance, the never-say-die-spirit and the support from the stands was nothing short of magnificent.
"We all left happy we had scored late on to get a deserved point from the game.
"As a result, we travelled down to Crystal Palace a few days later a little concerned at the physical and emotional impact the night might have had on our players.
'So once again it was great to come away from Selhurst Park with three enormously valuable points.
'It was a great effort and once again I want to say a huge thank you to our magnificent travelling support. It was great to give you something to smile about on the long journey home.
"We must not forget that our job is still to make sure we are a Premier League team next season."