The 19-year-old Russian, who is the reigning French Open champion, needed just 70 minutes to overcome the sixth-seeded Svitolina and record her first victory at the season-ending championship.
Andreeva initially appeared nervous but gradually found her rhythm, breaking Svitolina to establish a 4-2 advantage in the opening set before closing it out 6-4.
She carried that momentum into the second set, winning all six games to complete a convincing straight-sets victory.
Andreeva overcomes nerves to dominate
Despite her impressive performance, Andreeva admitted that she struggled with nerves before and during the opening stages of the match.
"I was so tight, I felt I was moving like a turtle," she said after the contest, expressing relief at eventually settling into the match and controlling her emotions.
The teenager's victory also represented revenge for her defeat to Svitolina at the Australian Open earlier in the 2026 season.
The result improved Andreeva's head-to-head record against the Ukrainian to 2-1.
Next challenge against Sabalenka
Andreeva will face Belarusian world-class opponent Aryna Sabalenka in her second round-robin match on Wednesday.
Sabalenka also opened her campaign with a victory, defeating Czech debutant Karolina Muchova 7-6(2), 7-5 in the other Group 2 encounter.
Both winners are now level at the top of the group with one victory each, while Svitolina and Muchova will look to recover in their remaining matches.
The top two players from each group will advance to the semi-finals as the battle for the 2026 WTA Finals title continues in Indian Wells.