Third-seed Andreeva, the 19-year-old reigning French Open champion, took control with a break to open up a 4-2 first-set lead and never looked back against her sixth-seeded opponent.
She duly held serve to close out the first set, and then demolished Svitolina in the second set, rattling off three service breaks to wrap up victory.
"I was so tight, I felt I was moving like a turtle," the teenager said afterwards. "I felt like I couldn't really hit the ball, so I'm just very happy to get through the match and control my emotions."
Andreeva will face Sabalenka in her second round-robin game on Wednesday.
This year's WTA Finals were scheduled to be held in Saudi Arabia but in July were moved to Indian Wells after US military strikes on Iran in February and the subsequent war in the region.