The former Everton midfielder has been hugely impressed by the start to the season under Moyes, crediting the manager for getting the best out of his players despite what he considers a lack of sufficient financial backing during the transfer window.
Everton have made an encouraging start to the Premier League campaign, with Moyes building a competitive side capable of challenging towards the top end of the table. Their performances have been particularly impressive given the circumstances surrounding the club before the season began.
"You can't help but be pleased with it. It's been a great start, and I know David Moyes will still be driving the standards even higher," Kilbane told Football Presse on behalf of Betway. "He'll be expecting more from the team and wanting more from the team.
"He's a great manager and a great person to get the best out of individuals. That's why Everton have been so successful this season. It's solely down to him.
"I know there's a lot of talent in the squad. Dewsbury-Hall and Garner are doing excellently in midfield, Thierno Barry is scoring goals up front, and everyone is playing their part.
"But David Moyes was under a lot of pressure at the start of the season. He wasn't backed financially or given the necessary funds to bring in the players that were needed. We all know that; it's visible to us all.
"But he knows what to do in these situations, and what he's done in the first 10 games or so has been phenomenal. There's nothing to suggest Everton can't continue what they've done this season, finish in the top six and maybe achieve something phenomenal by going beyond that."
Kilbane's praise comes at a time when Moyes' own future remains a significant talking point. The Scot is in the final year of his contract, while The Friedkin Group's decision to explore the sale of its controlling interest in Everton has added further uncertainty to the situation.
Moyes has already acknowledged that discussions over an extension have slowed amid the ownership developments, although he has maintained his focus on improving the team.
For Kilbane, however, the identity of the manager should be a straightforward decision, regardless of who eventually controls the club.
"Of course," he said when asked whether Everton should renew Moyes' contract.
"But I think that, regardless of the ownership at the club and whoever comes in, you're not going to get a better manager than David Moyes for what's needed right now.
"Given time, David Moyes has built teams like he did at Everton in the past, teams that go on to play great football. I've witnessed it; I've seen it.
"He hasn't achieved anything as yet, but given the state the club was in at the start of the season and the lack of investment that went into the team, everything is pointing towards an extension of his contract in some capacity."
Everton's recent results have strengthened the argument for keeping Moyes, with consecutive league victories over Newcastle United and Coventry City taking them into the international break in fifth place.
Kilbane believes the manager has already demonstrated why he remains such an important figure at Goodison Park's successor, the Hill Dickinson Stadium. While there is still work to do before Everton can realistically claim to have returned among the Premier League's leading clubs, the former Republic of Ireland international sees a manager capable of delivering that progress.
For him, the priority should be ensuring Moyes is given the opportunity and time to continue the work he has started.