The 21-year-old, who signed from Crystal Palace in January last year, stayed on loan at Selhurst Park until the end of the season before linking up with the United squad and their new manager.
Having managed just four appearances for Moyes' side, Zaha made a January loan switch to Cardiff City in a bid to play regular Premier League football, but has often found himself used as a substitute.
Zaha admits that the change in managers at United came as a bit of a shock, but says he did not get enough opportunities to prove himself.
"I signed for Sir Alex Ferguson and expected him to be the manager," he told The Sun.
"David Moyes arrived and obviously felt - in general - he had to play older players.
"For some games I was not even on the bench. That was tough, very frustrating. You need the manager to trust you and that was not the case.
"Raheem Sterling went through a tough period at Liverpool and there was talk of him going out on loan. But Brendan Rodgers trusted him and he was allowed to get that experience and pull through."
However, Zaha is desperate to prove himself at Old Trafford and says that he will do everything he can to impress upon his return.
"I hope to still have a future there. I will go in for pre-season and give my best. Had the manager stayed, I'm not sure that would be the case," he said.
"I have four years left on my contract at United. Had Moyes stayed I would have had to think about some other options.
"I will benefit from this tough year. Everything cannot go right all of the time."
The winger was also keen to put to bed some of the rumours which circulated as to why he had been cast aside at Old Trafford.
"Everyone was thinking 'how has he come from Palace to not play?' People thought I had a bad attitude," he said.
"There were rumours about Moyes' daughter from Twitter. It was weird. I was getting tortured for something I hadn't done. I hadn't even met her. That is why Twitter can be such a negative thing.
"Obviously the season did not go too well for Moyes. Some people have asked if I was happy he is no longer there but of course I am not.
"I don't want to see anyone get sacked."