Alcaraz, 19, had a testing time breaking through against his determined Dutch foe, who had won their only prior hard court meeting.
He didn't face a break point in the first set, nor could he convert either of his two.
A strong start in the tiebreaker proved the difference, Alcaraz locking up the set with a forehand winner on his second set point.
He rolled from there, breaking Griekspoor in the second game on the way to a 3-0 lead in the second set.
"It has been a really tough match," Alcaraz said of his 100th ATP Tour match win. "Tallon is playing great, I had to be really focused. Of course playing with a lot of wind like today is also tough.
"I had chances at the beginning of the first set, I didn't take it.
"The second set, I took my chances that I had in the beginning and thanks to that I was able to play more relaxed."
Alcaraz will face Britain's Jack Draper for a place in the quarter-finals after Draper overcame former world number one Andy Murray 7-6 (8/6), 6-2.
Women's world number one Swiatek, seeking to become the first woman since Martina Navratilova in 1990 and '91 to win back-to-back titles in the combined WTA and ATP Masters 1000 event, elevated her game when she had to for a 6-3, 7-6 (7/1) victory over Canadian Bianca Andreescu.
Andreescu, whose 2019 Indian Wells title proved the springboard to a breakout season that included a US Open crown, pushed Swiatek through punishing baseline rallies, trading six breaks of serve with the reigning French and US Open champion in the second set only to be overpowered in the tiebreaker.
"Bianca can change the rhythm pretty well on this surface, it can get tough," Swiatek said, but added she was "glad that I had a chance to play under pressure a little bit more and see how I'm going to cope with that."
Swiatek next faces another former US Open champion in Britain's Emma Raducanu, who earned an impressive 6-1, 2-6, 6-4 victory over big-hitting Brazilian Beatriz Haddad Maia.
Raducanu, who has battled nagging wrist trouble and tonsillitis in recent weeks, notched her best win by ranking since she captured the US Open two years ago.
Reigning Wimbledon champion Elena Rybakina of Kazakhstan punched her ticket to the fourth round with a 6-3, 7-5 victory over former champion Paula Badosa of Spain.
Rybakina, ranked 10th in the world after a runner-up finish to Aryna Sabalenka at the Australian Open in January, had lost her last three matches against Badosa -- her doubles partner this week.
Fourth-seeded Tunisian Ons Jabeur, playing her first tournament since missing Doha and Dubai in the wake of knee surgery, was beaten 7-6 (7/5), 6-4 by Czech Marketa Vondrousova -- the same woman who upset Jabeur in the second round of the Australian Open.
In another rematch of an Australian Open second-round clash, fifth-seeded Caroline Garcia beat Canadian Leylah Fernandez 6-4, 6-7 (5/7), 6-1.
- Fritz eases through -
In other men's action, defending champion Taylor Fritz cruised past Argentina's Sebastian Baez 6-1, 6-2.
Three-time Grand Slam champion Stan Wawrinka, 37, beat 19-year-old seventh seed Holger Rune 6-2, 6-7 (5/7), 7-5.
A former world number three now ranked 100th, Wawrinka avenged a first-round defeat to the Dane at the Paris Masters in November, where Rune saved three match points to launch his run to a first Masters title.
Murray, 35 and trying to grind his way back after hip replacement surgery, couldn't pull off a similar feat against his 21-year-old compatriot Draper.
Draper led the first set 3-1 but was broken at love as he served for the set at 5-4.
He had to save a set point with an ace but never trailed in the tiebreaker. Draper won the last four games to win his first meeting with his childhood hero.
"I've looked up to Andy since I was so young," Draper said. "I watched him win Wimbledon for the first time in 2013 ... He's a really special person, a great champion, great human being and I'm privileged to play against him on this court."
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