The Belgium international was given his big break at former club Manchester United by Moyes, making a big impression in 2013-14 before fading away and eventually moving on to Real Sociedad in the summer.
Moyes, who also signed the 22-year-old on loan while Sunderland boss last season, is now tasked with guiding West Ham back up the Premier League table after replacing Slaven Bilic in the London Stadium hotseat.
Januzaj believes that his former boss is a solid choice for the Hammers, telling Sky Sports News: "He's a good manager. Obviously it's been difficult for him too. I'm happy for him because it's a nice move and I hope everything goes well for him.
"He likes to play good football. It's not easy sometimes, it depends on the squad. People have to trust him because he's done it at Everton for a long time."
Januzaj has featured 10 times for Sociedad in La Liga and the Europa League since arriving in a £9m deal earlier this year.