In a game of few clear-cut chances, Wayne Rooney came as close as anyone with a long-range effort that brought a decent save out of Petr Cech.
Danny Welbeck screwed a decent headed opportunity wide in the second half, while Tom Cleverley had a penalty appeal turned down after his shot crashed into Frank Lampard.
"If anybody created the chances it was us," said Moyes.
"I picked an attacking team that was trying to score. We had boys who score goals and gave it a go.
"Maybe we lacked a real good pass through or a really top cross that might just have made the difference or it might have been an individual moment of brilliance that we have seen in the past."
And whilst most observers felt Lampard would have been exceptionally unfortunate to concede a spot-kick, Moyes felt his side had a genuine claim.
"There was a shout for a penalty," said Moyes.
"The referee had a good game but just afterwards he gave handball against Oscar for a lesser handball in the middle of the pitch.
"We could do with having it defined what is going to be given as a handball."