The world number one was repeatedly dared by Brown to dash in from the baseline and scramble for a ball dying on the grass - but after his comfortable 6-3 6-2 6-2 victory the Briton said he felt good and that he had moved well.
Murray gave as good as he got - and better - playing many of the same shots with more accuracy and control in a quickfire 96 minute palate-cleanser for the Centre Court crowd after the earlier epic three-setter won by compatriot Johanna Konta.
"If he has a problem with his hip, I don't want to play against him when his hip is good," Brown told reporters.
Brown, ranked 97th, was a potentially dangerous opponent, having beaten twice champion Rafa Nadal in the second round in 2015. But he showed the frailties of such high stakes hotshot tennis, racking up 28 unforced errors to Murray's five.
The home fans loved the show, though, temporarily forgetting their wildly partisan support for the Briton as Brown casually sliced and diced the ball over - and frequently into - the net.
In the first game of the third set, the German acknowledged a shout of "We love you Dustin" by raising his racket to the crowd before serving.
But for all Brown's flare, it was Murray who remained in control throughout and never looked like losing.
"He started very well and was coming up with great drop volleys and really going for the returns," Murray said.
"Once I got the break in the first set, I felt the momentum was with me, I was starting to see the shots he was going to play a little bit quicker and that allowed me to get to some of the drop volleys and also come up with some good passing shots."
The top seed's win was greeted by huge cheers from a crowd hoping to see the 30-year-old twice champion become the first British player to retain a grand slam title since Fred Perry in the 1930s.
After seeing off maverick Wimbledon debutant Alexander Bublik in round one and the clash with Brown, Murray faces another showman, Italian 28th seed Fabio Fognini, in round three.
"Certainly, the first two matches have been, difficult to come up with game plans, because you don't know exactly how the two guys are going to play," Murray said.
"I think against Fabio ... it's maybe easier to come up with a game plan because there will be a bit more structure and strategy in the match rather than just, reacting and sort of more kind of instinctive points."
(Editing by Ed Osmond)
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