Without work since he was sacked by Manchester United in April - just 10 months into a six-year deal - Moyes has taken the reins at struggling Sociedad.
Moyes has signed a contract until June 2016 with the San Sebastian-based club, who languish in 15th position after winning just two of their 11 league games so far.
Round, who was among the backroom staff to be relieved of their duties at Old Trafford following Moyes' dismissal, is confident the Scot is the right man to lead Sociedad's revival.
"First and foremost I'm really pleased to see him back in. He's a tremendous manager. He's got great honesty and integrity, and he's tactically excellent," Round said.
"He was a sad loss not to be in the game, you know. After Manchester United he decided to take a break, he needed a break, he wanted to re-energise.
"He'd been in the game a long time, he's been in management for over 13 or 14 years straight, and I just think he needed to get his energy back and his enthusiasm.
"He's a very talented manager, a very talented coach, and Sociedad is a club that should be in the top half of the Spanish league, that's for sure, then pushing towards the Champions League spots."
Round added: "He always said that one day he'd love to manage abroad, work in a new environment, a new culture and test himself on a different stage.
"So it's no surprise to me that he's chosen Spain and he's chosen a very good club. And it seems to me to be a two-way street – they've chosen him and he's chosen them, which is always a good mix.
"I think he's had a lot of interest in his services, from numerous different clubs, but I think he was waiting for the right club at the right time and the right approach.
"And speaking to him over the last couple of days, he was very taken with what they were offering – the philosophy of the club, to promote its younger players, to bring players through and to build on the success of the last couple of years.
"They believe they've got a very good side in the making. They've had to sell one or two of their stars to the big clubs, but they always believe there's the next one coming through the ranks."
