Spain's second-ranked Alcaraz, the US Open champion who can return to number one in the world with a third Masters 1000 crown, needed just one hour and 16 minutes to get past Australian qualifier Thanasi Kokkinakis 6-3, 6-3.
Women's number one Swiatek, winner of the French and US opens last year, was even more ruthlessly efficient in a 6-0, 6-1 triumph over American Claire Liu.
Alcaraz lost just one point on his serve in the first set and did not face a break point in the match as he lined up a meeting with Tallon Griekspoor, a 7-6 (7/3), 7-6 (7/4) winner over Argentina's Guido Pella.
"Well, I surprised myself today with my serve," the effervescent 19-year-old said. "I serve really well in the first set and in the second set as well."
Early breaks in each set were too much for Kokkinakis to overcome in the face of Alcaraz's speed and power.
"All that I'm looking for in every match is to enjoy and to play relaxed," Alcaraz said. "I think that's the most important part of my game, to stay relaxed and go for it and not to think about the mistakes, the points, the rounds.
"Just live the moment, play the match, and go for it," he added.
Alcaraz was able to do just that as there was no sign of the right hamstring injury that forced him out of the ATP tournament in Acapulco prior to Indian Wells.
He had missed the Australian Open with similar trouble, but returned to win in Buenos Aires last month before falling in the final at Rio de Janeiro -- where he was hindered by the injury.
"It seems today that I move really well," Alcaraz said. "I think I'm ready and totally recovered from the injury."
Defending men's champion Taylor Fritz, who toppled Spanish superstar Rafael Nadal in last year's final, had to work harder to get through his second-round opener, a 4-6, 6-4, 6-3 victory over 41st-ranked Ben Shelton.
Seventh-seeded Dane Holger Rune and eighth-seeded Canadian Felix Auger-Aliassime advanced.
Rune beat American Mackenzie McDonald 7-5, 6-3 while Auger-Aliassime beat Spain's Pedro Martinez 7-6 (7/5), 6-4.
- Murray advances -
Former world number one Andy Murray, grinding his way back after hip replacement surgery in 2019, defeated lucky loser Radu Albot 6-4, 6-3 -- the Briton's first straight-sets win in a year that has seen him pull off a string of unlikely marathon victories that included a five-set epic lasting five hours and 45 minutes in the second round of the Australian Open.
Swiatek, meanwhile, needed just 65 minutes to dispatch Liu as she launched her bid to become just the second woman -- after Martina Navratilova in 1990 and '91 -- to win back-to-back Indian Wells titles.
Neither Liu nor the breezy conditions on Stadium Court troubled Swiatek, who won 25 of 32 first-set points.
Liu did manage to save a match point as she finally held serve in the penultimate game, slipping a forehand passing shot under Swiatek's guard on game point.
It brought a relieved smile to the American's face, but Swiatek closed it out in the next game.
"For sure Claire used that moment when I didn't play as aggressive that I should be, but I'm pretty happy that I was able to close it out pretty fast," Swiatek said.
She can expect a tougher challenge when she faces 2019 Indian Wells and US Open winner Bianca Andreescu, who defeated American Peyton Stearns 4-6, 6-4, 6-3.
Women's fourth seed Ons Jabeur of Tunisia made a winning return after being sidelined in the wake of a minor knee operation, beating Poland's Magdalena Frech 4-6, 6-4, 6-1.
Fifth-seeded Caroline Garcia overcame Hungarian Dalma Galfi 6-1, 6-7 (4/7), 6-4 and Wimbledon champion Elena Rybakina, runner-up at the Australian Open in January, rallied from 4-1 down in the second set to beat former Aussie Open winner Sofia Kenin 7-6 (8/6), 7-6 (7/5).
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