The Brazilian full-back will join the Welsh club on a permanent deal, while the England Under-21 international winger will move on loan until the end of the season
The 23-year-old Brazilian has made just three appearances for United this season and was sent off in the club’s FA Cup third-round exit to Swansea City earlier this month.
"We're delighted for Fabio because he has been a great servant," Moyes told MUTV.
"His brother is doing a great job for us here as well.
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“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.”
Goal exclusively revealed on January 17 that Zaha had stormed out of training after learning United officials were not willing to send him out on loan this month.
But Moyes now backs Zaha’s temporary switch to Cardiff and says Solskjaer is the right manager to get the best out of the winger.
"With Wilfried Zaha, I'd said to him during the summer that we wanted to get him games. If he played, then great, but if he couldn't, we'd get him out on loan,” Moyes said.
“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."