The midfielder was Strachan's most expensive signing during his four years as Celtic boss, arriving from Hibs in 2007 for £4.4m.
Brown has won 29 caps since making his Scotland debut for Walter Smith seven years ago, and is in no doubt the right man has been appointed to pick up the pieces post-Craig Levein.
Brown said: "Good to see Gordon getting the Scotland job. We couldn't ask for a better manager."
That support is expected to be echoed by Brown's Scotland team-mates, and some who have been frozen out in the latter stages of Levein's reign.
Strachan has warned it will take everyone pulling in the same direction if the national team's fortunes are to be turned around.
And he made it clear that he offers no cure-all panacea, or a quick fix.
Strachan said: "There will be rough times, but I know there'll be good times, too.
"If we work together as a group and a set of fans, I know for a fact we'll be successful.
"I can't ask for anything more than we're doing at the moment."
The immediate aim is to get a win against Estonia in the friendly at Pittodrie on February 6 to lift morale ahead of the World Cup double header against Wales and Serbia in March.
Strachan concedes that recovering from the poor start made to the group – two points from four games – leaves no margin for error.
But he refuses to call it mission impossible and wants to establish a winning run and mentality.
"My philosophy is to win games of football, like Manchester United," he said.
"We still have to collect as many points as we can, and it's disrespectful to say we'll use the [World Cup] games just to improve. We'll give it a go.
"What I know is that these guys, for all they might get criticised, are the best at what they do.
"They play for Scotland and we have to find a system that suits these players to win games."
Strachan enjoyed being part of the World Cup finals in Spain 1982 and Mexico four years later, and wants the current group of players, and the fans, to have the chance to experience this for themselves.
He said: "I really want the squad and the staff to give something back to the country and the fans who support us because the fans are probably more famous than the squad now.
"We want them turn up for a major finals competition."
During his extensive discussions with the SFA, who made him their No.1 target as soon as Levein was removed in November, Strachan constantly studied the form of players he might consider for the Estonia match.
But he is not expected to make many changes in personnel, just the system and mind set.
Just over three weeks before his 56th birthday, Strachan is confident this is the right team to take on such a challenge.
His contract will run through this World Cup campaign and to the European Championship which follows in 2016.
But he is fully aware of the need to get off to winning start and validate his credentials as the manager the country can trust to restore pride.
He said: "At this time in my life, I can deal with this. All my experiences have prepared me for this moment.
"The time is right for me to be able to take a job like this.
"I have the experience, I like working with people, I like dealing with people and I like trying to make people better.
"Every club you go to, there's always challenges. But this is the first time the challenges are in front of a nation.
"If you can be successful, you can make a nation happy and make a nation proud.
"That's the priority for me, to be able to do that with help from other people."