Mr Brown described him as "a great football ambassador" who would never be forgotten.
Jardine, a 38-time capped Scotland international, was diagnosed with cancer 18 months ago. He died on Thursday night surrounded by family.
Mr Brown said: "I met Sandy Jardine on a number of occasions when he led football tributes to my friend Jim Baxter.
"Sandy was a brilliant right back, a world-respected Scottish internationalist and a great football ambassador who will never be forgotten. My thoughts are with his family."
Celtic boss Neil Lennon also paid tribute, describing Jardine as "one of the best Scottish defenders of his generation".
He said: "I would like to send my condolences to his family. Sandy was one of those rare people who transcended the Old Firm and he was very well respected within the fraternity here (at Celtic).
"He was always a gentleman, a huge Rangers man, obviously, but very well respected by everyone. It is very sad, 65 is not that old."