Hamilton clinched his sixth F1 drivers' championship in Texas last weekend, moving just one behind the all-time record of seven held by Michael Schumacher.
Brawn, who worked with Hamilton in 2013 at Mercedes, admits that he was "nervous" when he first started operating with the Briton.
"I think the best compliment I can pay Lewis today is to say that he has found a unique way of being a great champion," Brawn wrote in his post-race column.
"I have to admit that when I started working with him back in 2013 I was slightly nervous about his approach to being a professional racing driver.
"For me, the benchmark was Michael Schumacher, a man totally dedicated to chasing success and at the same time very keen to guard his privacy and that of his family.
"Even then, Lewis was demonstrating a truly eclectic approach to his career and whilst totally professional he preferred a lifestyle that was definitely less monastic than Michael’s."
Brawn applauded the efforts of Mercedes over the last handful of years, who Brawn says deserve credit for continuing to the success of Hamilton.
"At Mercedes, Lewis has found the right support in an environment that is stimulating technically, but that at the same time allows him the freedom he absolutely needs.
"I therefore congratulate Toto Wolff and the whole team, as they’ve not only given him an amazing car year after year but they’ve allowed Lewis to develop and grow to become one of the greatest drivers in the history of Formula 1, certainly the strongest of the last 15 years."
Bahrain International Circuit - Winter testing