"I am really happy I can win the first match (as No. 1)," said Halep, the first Romanian woman to be top-ranked. "It takes the pressure off."
"I just kind of broke her down a little bit," Wozniacki said. "Then she lost the belief or she tried to go a little bit more for it. She just couldn't find a break or find a way to break me."
All three players making their WTA Finals debut -- Garcia, Svitolina, Jelena Ostapenko -- lost their first round-robin matches in straight sets.
Halep faced six break points and lost only one when she conceded her serve with a double fault in the sixth game.
Garcia, making her WTA Finals debut, was broken twice in each set.
Halep, making her fourth consecutive appearance in the WTA Finals, was a finalist here in 2014. She lost to No. 1 Serena Williams, which marked the last time the top-ranked player won the season-ending event.
Svitolina, the first Ukrainian woman to play at the WTA Finals, arrived in Singapore having won a tour-leading five titles this season.
"Today was a terrible day for me," Svitolina said. "Caroline played really, really good. It was probably my worse match I've played in a couple of years."
Halep and Wozniacki stand at 1-0, while Garcia and Svitolina are 0-1 in Red Group action.