Bony, goalless so far since his season-long loan from Manchester City, was denied a breakthrough goal by keeper Jordan Pickford after he stabbed a shot goal wards from 16 yards.
The assistant referee correctly ruled out Arnautovic’s effort in the 67th minute after he deflected home Charlie Adam’s mis-hit shot.
Moyes said: “It was a nightmare start. I didn’t think we played as bad as 2-0 in the first half suggested.
“I thought we played better in the first half.
“They were two poor goals to concede.
“If we could have gone in at half-time at 1-0, I think we had enough in the game to have a chance.”