Distin headed home from a Leighton Baines cross, but referee Michael Oliver penalized Victor Anichebe for a foul on Reds goalkeeper Pepe Reina.
"It was a goal," Moyes told Sky Sports. "The goalkeeper runs into our player so the goal should stand.
"If we run into the goalkeeper then give a foul for that, but he ran into us, so it was a goal. The referee said he gave the foul on Victor Anichebe for impeding the goalkeeper."
Moyes was reasonably satisfied with his side's display at Anfield but was felt that events conspired against his team away from home. The draw ensures the sixth-place Toffees retain a five-point lead over Liverpool with two games left to play.
"I thought we did what we had to do," Moyes said. "We didn't make as many chances as we would have liked.
"We found it hard to get a lot going for us, things weren't going our way. That's what happens when you come to Anfield, you very rarely get good decisions.
"We are still in there. We picked up a point, we've had a really difficult run-in and we will keep pushing."