An energized Shelton came out firing and crushed an emphatic overhead to break serve en route to grabbing the first set as an upset appeared in the works.
But the fourth-seeded Fritz converted the first break opportunity he saw on set point when Shelton dumped a forehand into the net to level the match at a set apiece.
In the final game of the match, Fritz showed that he is more than a big server, hitting a brilliant lob and deft drop shot to set up match point, which he converted when Shelton's attempt to block back a ferocious forehand sailed long.
Take a look at the match summary here.
"He's a really good player, and I expected him to come out serving bombs and playing well because he had nothing to lose," Fritz said of Shelton.
"I had to regroup and problem-solve. He's an extremely tough first match to play so I'm happy to get through it."
Reigning US Open champion Alcaraz never faced a break point against Australia's Kokkinakis as the Spanish teenager cruised to a stress-free win in his first hard-court match of the year. It finished 6-3, 6-3.
Andy Murray (35) enjoyed a rare straight-sets win over tournament lucky loser Radu Albot (33) of Moldova 6-4 6-3 to set up an all-British third-round showdown with Jack Draper (21).
The veteran Murray said he expects a great match against a player he has mentored. Draper beat countryman Dan Evans 6-4 6-2 earlier in the day.
"I think he'll be a top player for a long time," Murray told reporters.
In other men's results, Italian 11th seed Jannik Sinner (21) beat Frenchman Richard Gasquet (36) 6-3 7-6(2), eighth seed Felix Auger Aliassime (22) of Canada was a 7-6(5) 6-4 winner over Spain's Pedro Martinez (25) and Swiss Stan Wawrinka (37) beat 26th seed Serb Miomir Kecmanovic (23) 7-6(8) 6-4.
SWIATEK CRUSHES LIU
On the women's side, world number one and top seed Iga Swiatek (21) got her title defence off to a dominant start with a 6-0 6-1 thrashing of Claire Liu (22).
"Tactically and tennis-wise, everything was on point, so I'm pretty happy with my game," said Swiatek, who next faces 2019 US Open champion Bianca Andreescu (22) of Canada.
Emma Raducanu (20) knocked out Polish 20th seed Magda Linette (31) 7-6(3) 6-2 to notch her biggest win since triumphing at the US Open in 2021.
The Briton fell into a 4-1 first-set hole but rallied from there, dictating points with her forehand to push Linette deep into the court.
She sealed the win by whipping a forehand winner down the line to bring the fans to their feet on a warm and windy day in the Southern California desert.
The win marked Raducanu's first back-to-back victories since September and is a hopeful sign that she may be moving beyond the multiple health issues that have beset her of late.
She suffered an ankle injury in the lead-up to January's Australian Open, tonsillitis that forced her to withdraw ahead of last week's Austin Open and a flaring up of the wrist injury, which had cut short her 2022 season and required attention from a physiotherapist on Saturday.
"I'd say I feel more positive about my game today," Raducanu told reporters.
"The first day I didn't really know what was going on. Today I felt pretty decent out there, especially with the conditions. It was really windy and I did well to overcome it."
She will next face Brazil's Beatriz Haddad Maia (26), who was a 5-7 7-6(4) 6-3 winner over Czech Katerina Siniakova (26).
The tournament's 2021 champion, Paula Badosa (25), scored a 6-2 7-5 win over fellow Spaniard Nuria Parrizas-Diaz (31) in the night session.
Elsewhere, Czech Karolina Muchova (26) scored an impressive 7-6(1) 6-3 win in her tournament debut over two-time champion Victoria Azarenka (33) of Belarus and Czech Marketa Vondrousova (23) crushed her compatriot, the 28th-seeded Marie Bouzkova (24), 6-1 6-1
