Stubbs served as Everton captain under Moyes a decade ago.
Moyes was named as Sam Allardyce's successor at the Stadium of Light at the end of last week after Allardyce left Wearside to take over from Roy Hodgson in the England role.
That appointment marks a return to the Premier League after two years for Moyes, who has been away from the UK since being sacked by Manchester United in April 2014.
Though his time at Old Trafford did not go to plan, he was the man tasked with the virtually impossible role of succeeding Sir Alex Ferguson's 26-year reign, and Sunderland fans will no doubt be looking back to his time at Everton.
During a decade at Goodison Park, Moyes turned Everton from a lower-mid-table club into European challengers, and even led the club to break into the top four of the Premier League and consequently qualify for the Champions League.
Moyes' captain in his first few years at the Toffees, Stubbs believes that his former boss has what it takes to bring stability and long-term improvement to the Black Cats in the same manner that he achieved on Merseyside.
Stubbs told the Mirror: "It’s a good appointment for Sunderland and it’s great that he is back in the Premier League.
"He brings stability because they will have been rocked by Big Sam leaving but they have moved quickly. It is a similar situation to when he took over at Everton, and when he left they were involved in the top six every season.
"Sunderland is a really good fit to where he is now. He will want to make the Stadium of Light a fortress and his teams will be very hard to beat, which is the David Moyes style."
Moyes will certainly relish the opportunity to build his own project again as he did at Everton, and also to restore his own reputation after his failure at Old Trafford.
His first task will be to ensure that builds upon the work done by Allardyce and begins to guide Sunderland away from their annual relegation battle and up into mid-table.