Swiatek, playing at a Grand Slam tournament for the first time since linking up with Rafael Nadal's former coach Francisco Roig, showed flashes of her best tennis in a 6-1, 6-2 victory.
The Pole is looking to regain the trophy she won at Roland Garros in three successive years from 2022-24 before losing to Aryna Sabalenka in the semi-finals 12 months ago.
Swiatek will take on Czech Sara Bejlek for a place in the last 32 on Wednesday.
The third seed is among the favourites for the title after finding some form by reaching the Italian Open semi-finals earlier this month.
Swiatek led 6-1, 3-0, before 17-year-old wildcard Jones battled for respectability, holding serve for the first time in the fifth game of the second set.
An upset was never on the cards, though, and Swiatek wrapped up victory after just an hour on Court Philippe Chatrier.
"I'm just really happy to play on this court; the first matches are for getting used to conditions, I'm happy the way I played today, tactically," said the 24-year-old, who has lost only two matches at Roland Garros in six years.
Her pedigree, however, comes with added pressure.
"Nothing comes easy. I feel with more titles it feels harder because everyone expects you to play perfectly, so you have to stay grounded and not take anything for granted."