Zaha is set to link up with Cardiff on loan for the rest of the season after struggling to make an impact since Moyes' appointment as United boss.
The England international has made just two substitute appearances in the Premier League and had been linked with several teams, including his former club Crystal Palace.
However, the 21-year-old is close to finalising a move to Cardiff, who are managed by former United striker Ole Gunnar Solskjaer.
"With Wilfried Zaha, I'd said to him during the summer that we wanted to get him games," Moyes told MUTV. "If he played, then great, but if he couldn't, we'd get him out on loan.
"We had one or two offers for him but just felt Cardiff, at this moment in time, might be the right club.
"With Ole going there as well, he knows how the club works here and exactly what we do with the players so hopefully it will be a good move for Wilf."
Full-back Fabio leaves after making just 56 appearances for United since pitching up at Old Trafford with his brother Rafael in 2008.
Though Rafael has established himself as a first-team regular, Fabio has been unable to impress as much and spent last season on loan at Queens Park Rangers.
The 23-year-old has appeared just three times under Moyes and was sent off on his final outing in the FA Cup third-round defeat by Swansea City earlier this month.
Moyes said: "We're delighted for Fabio because he has been a great servant. His brother is doing a great job for us here as well.
"We wish him well. He had one or two options but, in the end, we felt we couldn't get him enough games.
"I said to the players, if you feel I'm not getting you enough games, come and see me. Fabio felt that so he's got a move to Cardiff."