The former Manchester United defender made the breakthrough in the 52nd minute at Goodison Park.
Neville featured in all but three minutes of the game and Moyes said: "He was up all night with sickness and it was only at 11am he was cleared to play.
"That is why we took him off. He wasn't well at all.
"You have to give him credit. He played well in the circumstances.
"As for the goalkeeper, Tim has gone and attacked the ball.
"The goalkeeper hit Tim, not the other way around and it flies into far end of net. We are delighted."
Moyes told his side at half time to up the tempo after they were left frustrated by Bolton and they responded.
Cahill's goal - his seventh in 13 games in all competitions - came after Thomas Gravesen had been brought down by Gavin McCann.
Referee Rob Styles played the advantage and Bolton were caught cold.
The ball broke to Joleon Lescott and he played it back into the middle where Cahill made no mistake.
Moyes said: "When I saw Tommy go down, I thought it was a free-kick on the edge of box.
"But the first rule you are taught is to play to the whistle and we kept going.
"It was a good advantage from the referee and we are pleased with the victory. We showed no reaction to that defeat by United.
"We just needed to speed up the play and move the ball a bit faster and we did that in the second half."