Tierney made the move from Parkhead to the Gunners for £25 million ($32m) in the summer but has struggled to make an immediate impression for Unai Emery’s side due to injury.
Brown, though, says that once his former club-mate is up to speed, Arsenal will appreciate what a good player they have on their hands.
“He’s a fantastic full-back, he’s got everything you could ask for,” Brown told TalkSPORT. “He’s got pace, he’s got strength, he’s got that drive inside him that wants to win.
“He had that at Celtic, he showed his qualities and that’s why Arsenal wanted him.
“I was sad to see him go because he’s a great player and a great friend as well, but it was the right time.
“He will kick on and show everyone what a quality player he is.”
The 22-year-old has been restricted to just a couple of Premier League appearances since making the move south, featuring in successive home matches as Arsenal drew at home with Crystal Palace and Wolves.
He has, however, made a greater impression in the Europa League, where Emery has trusted him as the starting left-back in three successive matches. Tierney has played a significant role in victories over Standard Liege and Vitoria Guimaraes, in which he fashioned goals in both games.
Despite his relatively tender years, he is a player of surprising experience, having featured nine times for Celtic in the Champions League as well as 97 times in the Scottish Premiership.
He will not feature in Scotland’s forthcoming Euro 2020 qualifiers against Cyprus and Kazakhstan at the request of Arsenal, who wish to rid him of his ongoing fitness issues.
His absence has attracted criticism but he has been defended by James Forrest, another former Celtic cohort.
“He is certainly committed to Scotland,” said Forrest. “He played for Scotland at an early age and he has already captained Scotland as well.
“KT loves playing for his country but he was out for a while and is just getting back in the fold at Arsenal. I think Scotland and Arsenal will both want what is best for KT.
“The main thing for him is just to keep doing what he’s doing at Arsenal and be ready for the Nations League play-off games in March.”