Zaha signed a new five-year contract at Selhurst Park in August 2018 but was heavily linked with Arsenal and Everton and handed in a transfer request this week.
However, Hodgson insists Palace's business for this window is finished and that means keeping hold of the club's star attacking asset.
The former England manager acknowledged Zaha's desire to leave and confirmed he sent the player home from training.
However, he does not expect any lingering problems with the winger heading into a Premier League season where Palace are many people's tip for relegation
"Yes, he's staying, we are pleased about that. He's an important player for the club," Hodgson told Sky Sports News. "We're looking forward to him doing this season what he's done for the last two seasons.
"He wasn't really in the right frame of mind today, so I sent him home today which is understandable.
"It's been well documented that he's wanted to leave but it has not worked out for him in that respect.
"He's a professional and has got a good contract with us. He's very much a player we respect and I believe he respects us.
"We expect him to come back and do what he does and we're looking forward to a season with him again. Everyone here has a good relationship with him. There's not a problem with that.
"His beef is with the chairman and owners of the club because he's wanted to leave and they haven't received the offer that they think is sufficient to allow him to leave.
"He has to come to terms with that, when you sign long-term contracts you are expected to honour them and we expect him to do that.
"I don't have any issues with him. Today wasn't the right day so I'm looking forward to seeing him tomorrow and hopefully he'll have come to terms with the fact he's got another season with Crystal Palace."
Palace lost right-back Aaron Wan-Bissaka to Manchester United earlier in the transfer window but wrapped up their business by bringing in midfielder James McCarthy from Everton.
"It's good to have it done so we're not sweating on any last-minute things," Hodgson added.