The 19-year-old singer brought the dog home after finding it abandoned outside a budget store.
Cyrus tweeted: "He was left in a box in front of Walmart. I don't understand how people can be so cruel. That's why we named him Happy."
The animal - which she describes as a possible Rottweiler-beagle mix - has gone "from cardboard" to Maison Martin Margiela, according to the teenage star.
Cyrus shared photos of her new pet lying on a leather dog bed and accompanying her to a hair salon.
Her boyfriend - Hunger Games actor Liam Hemsworth - was also photographed on Tuesday picking up bags of pet food and a new dog bed.
Cyrus already owns three dogs named Lila, Floyd and Ziggy.
She recently encouraged friend Khloe Kardashian to adopt a dog in order to keep her company when she moved to Dallas, Texas with husband Lamar Odom.