The Toffees will provide the first opponents for new boss Mauricio Pochettino when they travel to the south coast for Monday's Barclays Premier League clash.
Pochettino, formerly in charge of Spanish side Espanyol, took over the Saints hotseat on Friday following the sudden sacking of Nigel Adkins.
The former Scunthorpe boss had achieved back-to-back promotions and appeared to have turned things around after a shaky start to life in the Premier League but that was not enough for ambitious chairman Nicola Cortese.
Adkins was hugely popular and the decision will have shocked fans and players alike but Moyes does not think it will make his side's task easier.
The Scot said: "It's a really difficult one to judge because I actually think Southampton have been playing quite well. I'm not quite sure what effect it will have on the players but when you get a new manager you get on with it.
"I don't think players mope around too long about the manager, and I mean that with real respect for Nigel, but I think players just get on with their job. I would expect them to do that and it will be difficult.
"I don't know what the atmosphere will be like, I would have expected the supporters to have probably been in Nigel's camp because of the work he'd done there."