The Belarusian turned up the heat in sweltering conditions, winning the opening five games as a quick finish looked on the cards.
Branstine got warm applause from the crowd as when she won her first game of the contest but Sabalenka eventually wrapped up the first set.
The second set was a more intense affair as Branstine gave a better account of her herself as the score moved to 5-5, before Sabalenka sensed her chance and broke her opponent's serve.
"I'm super happy to be back and healthy and able to compete this year at this beautiful tournament," said Sabalenka, a three-time Grand Slam champion and twice Wimbledon semi-finalist.
"In the first set she didn't serve that great but in the second she was serving every serve at 120mph, which was crazy. It was a tough second set and I'm super happy with the win."
She will face either last year's surprise quarter-finalist Lulu Sun of New Zealand or Czech Marie Bouzkova in the second round.