The footballing world will be focusing on Moyes when he takes his place in the Manchester United dug-out against Swansea, to begin in earnest the unenviable task of succeeding Sir Alex Ferguson in the managerial hotseat.
But trips to south Wales have held few fears for Moyes since Swansea joined the Barclays Premier League ranks in 2011.
His Everton teams enjoyed comfortable victories over the Swans by margins of 2-0 and 3-0, at a venue where few top-flight sides have found it easy going.
Midfielder Britton played in both those games and, with pundits tipping Swansea to upset the defending champions, is hopeful of turning the tables on the new United boss.
"He seemed to get his tactics spot on, but hopefully come Saturday that won't be the case.
"It will be all eyes on Manchester United to see how things go now Sir Alex Ferguson has left.
"It is a great occasion to be involved in, it's a new era at that football club and the eyes of the whole world will be on us. If we can cause an upset and get a result that would be great."
But whatever the outcome Saturday evening, Britton believes United have got the right man for the job.
"I think he can continue their success," he said.
"If you look at the job he did at Everton over the last 10 or 11 years, he did a fantastic job without too much money to spend and they have regularly finished in the top 10, just missing out on the Champions League.
"I think he is a very good appointment, we know how good a manager he is first hand, and I think he will be successful there."
One of the major question marks over Saturday's game is whether Wayne Rooney will feature for the visitors.
The striker, who is keen on a move to Chelsea, made his first appearance of the summer for England against Scotland on Wednesday after hamstring and shoulder problems, and national team boss Roy Hodgson reported no ill-effects.
Moyes has said he will assess Rooney'sfitness before making a final decision on his availability, and Britton admits he would be happy to not see the 27-year-old's name on the United teamsheet.
He said: "I don't know if we will face him. I watched the England-Scotland game and he played about 60 minutes.
"For him it has probably not been the best of pre-seasons, he has not been playing too much football.
"Being a player and a fan you watch what is going on and wonder what is going to happen with the talk of him leaving.
"But all I know is Wayne Rooney is a fantastic footballer, I have been lucky to play against him a few times and in my role I come directly up against him.
"I know if I am not on top of my game and give him a couple of yards of space he can get a shot away or a pass, so from my point of view, and Swansea's point of view, we hope he does not play."