Manchester United endured a harrowing campaign under Moyes last term as they finished seventh - their lowest-ever position in Premier League history.
The Red Devils slipped to 11 defeats under Moyes, before losing one more during Ryan Giggs' stint as interim boss after the Scot was sacked in April.
Zaha, who had been signed by Sir Alex Ferguson in January 2013 from Crystal Palace, made just three appearances for United last season, and speaking on the club's tour of the United States, the England Under-21 winger claimed that his face didn't fit under Moyes.
He told The Guardian: “To be honest, if you are not the manager’s cup of tea you can’t do anything about it.
“There was nothing I could do. It was definitely not my attitude. The manager had his way of playing and maybe I just didn’t suit it. There was so much pressure on him, so I can’t really blame him. I could have got a better chance but coming to United brings a load of pressure.”
The 21-year-old also tellingly offered an insight into what could be viewed as the fundamental differences between Moyes and new boss Louis van Gaal.
"[Van Gaal] won’t beat about the bush or not tell you anything," he said. "He will just tell you and that helps you instead of isolating you or just not telling you. He will let you know exactly where you are and what you are doing wrong.”
Zaha has so far been afforded a chance to prove himself in pre-season, albeit in a slightly unusual position as a striker, and the youngster said he was more than ready to adapt to Van Gaal's tactical tweaks.
“I am used to being a winger but if a manager wants me to adapt to a new position, then I will have to – and I don’t mind," he added. "To be honest, it’s not as much running as being a winger, so I will take it.
“At United there are quality players and if you get your chance you have to take it. Any chance I get I will try to make the most of it. You get ups and downs. The season before at Palace was a great season but last season was a bad one. I start off fresh this time.”
Zaha spent the second half of the campaign on loan at Cardiff City.