Still smarting from an unsettling French Open exit, the world number one was laser-focused on Centre Court as she won the opening four games against the lowly ranked Kostovic.
She was eventually made to work and there was a minor blip when she dropped serve at 5-2 in the second set but it was only a temporary delay as she avoided any possibility of the sort of collapse she suffered in Paris against Diana Shnaider.
"I'm super excited to be back," she told a crowd that included former England football captain Sir David Beckham.
"I was happy I could close this match in straight sets. For the first match I feel pretty good, maybe eight out of 10."
Sabalenka, a losing semi-finalist on her last three visits to Wimbledon, will take on American McCartney Kessler next.