There were some nervy moments for the world number one when he was seen limping between points, but he came through to claim a 6-3 6-2 6-2 victory.
Murray's fellow Brit Johanna Konta took over three hours to seal a spot in the next phase of the Grand Slam following a gruelling test against Donna Vekic, but her male counterpart needed just one hour and 38 minutes to get over the line.
The Scotsman showed the first signs of attack when he created a break point in Brown's second service game, but the German held firm.
Murray eventually found the breakthrough before successfully serving for the set, and he continued his dominance with a double break in the second.
The two-time Wimbledon champion was in a comfortable position on the scoreboard when the third set began, and once again he claimed two successive breaks before wrapping up the match.