The race carries a prize of A$10.2 million ($7.04 million; £5.3 million). It is Australia's third-richest race but is regarded as the most prestigious.
One runner, Asterix, was scratched earlier on Tuesday after a stable worker was found to have given it the medication penetrene that morning.
The victory completed a notable chapter for a horse that left the royal stable and arrived in Australia as a leading chance.
The horse was bought from King Charles earlier this year for an undisclosed sum and had been considered an early favourite.
Brown drove Point of Law from the middle of the pack to the lead with a few hundred metres remaining.
Last year the Melbourne Cup attracted a record A$382.5 million in bets.
Brown said of the winner:"He's in the best of hands with Ciaron Maher.". Protesters held signs with slogans including"horse racing kills","don't bet on cruelty"and"no more whips".