The two clubs meet in the first official fixture at the Premier League's newest venue today (2pm).
Tempers flared last time they did battle with Fabian Hurzeler being told he needed to learn a lesson in dignity.
Composure and concentration could be key as the Toffees look to impress at Hill Dickinson Stadium.
Roma enjoyed something of a smash-and-grab 1-0 friendly win at the new venue recently.
But the serious business starts today with Everton keen to bounce back from a 1-0 defeat at Leeds last Monday.
Albion were also the visitors when they kicked off their final season at much-loved Goodison Park a year ago.
Hurzeler recalls how his side weathered a torrid start and secured a 3-0 win in his first Premier League game as a head coach.
The Albion boss said: “I experienced during the first game I had in Everton, there was a special atmosphere, a loud atmosphere, and we expect the same this weekend.
“Everton supporters are known for being very emotional, they’re known for being great supporters, for being great supporters for their team, so we are looking forward for this atmosphere.
“We are looking forward to seeing the new stadium, but we are not there to be that nice club. We are there to win points and that’s our goal.”
Hill Dickinson has a capacity of 53,000 and Moyes wants to send the vast majority home happy.
Of all the clubs to have moved into a new permanent ground in the Premier League era, only Derby County in 1997-98 enjoyed a better record in their first year than the previous season at their old one.
But the Toffees were showing signs of progress last season even before their Goodison farewell.
Everton boss Moyes said: “Everybody knows that a new stadium is never that easy, but it’s something we’re all really excited about having now.
“I hope the crowd are really excited by it as well. We have to give them something to shout about and hopefully we can do that.
“There would be no point in us moving from Goodison if they weren’t going to come with the same vigour and the same support.
“I’m guessing after this week there will be less talk about the new stadium. For me it is now talking about the rebuild of the team and going forward.
“We’ve got the stadium now, now we’ve got to find a way of building a really strong Everton.”
Everton ground out a 1-0 success in last term's return fixture at the Amex which had Merseyside fans and media enthusing over how veteran Moyes had outwitted his young opposite number.
Hurzeler, who had himself out-manoeuvred vastly-experienced Sean Dyche at Goodison in August, responded: “You can write it like this, but that was not the key for this game.”
The Liverpool Echo commented after Everton's win: “Becoming a long-term success in the world’s toughest domestic division also entails having to deal with defeats in a dignified manner and that’s a lesson that Hurzeler and some others at Albion seemingly still have to learn.”