The highly rated Spanish teenager eventually downed the German wildcard 3-6 6-1 7-6 (7-3).
Kuhn, ranked 259 in the world, was aggressive early on against Alcaraz, and was rewarded with the first set.
But the 19-year-old – playing as top seed in an ATP Tour event for the first time – rallied to ease through the second set, before being made to work much harder to clinch the win on a tie-break.
Alcaraz will now play Filip Krajinovic in the second round after the Serbian also won in a third-set tie-break against Sebastian Baez 6-1 4-6 7-6 (8-6).
Three seeded players crashed out on Tuesday in straight sets, with third favourite Diego Schwartzman losing against Emil Ruusuvuori 7-5 6-4, sixth seed Nikoloz Basilashvili beaten by Aslan Karatsev 6-4 6-0, and eighth seed Holger Rune going down 7-6 (10-8) 7-5 to Tallon Griekspoor.
Fourth seed Pablo Carreno Busta eased through against Luca Nardi 6-2 6-1, while there were also wins for Fabio Fognini, who sealed his 400th career victory, as well as Daniel Elahi Galan, Borna Coric and Francisco Cerundolo.
At the Swiss Open in Gstaad, sixth seed Cristian Garin lost 6-3 6-4 to Yannick Hanfmann and seventh favourite Hugo Gaston fell to Dominic Thiem despite winning the first set, losing 1-6 6-1 7-6 (9-7).
Elsewhere, Frenchman Richard Gasquet beat Roberto Carballes Baena 7-5 6-4, while his compatriot Benoit Paire retired hurt when a set and a break down against Elias Ymer.
A tight game between Swiss pair Dominic Stricker and Marc-Andrea Huesler saw the former prevail 6-7 (2-7) 7-6 (7-2) 6-4, and an all-Spanish affair was similarly close as Jaume Munar defeated Bernabe Zapata Miralles 6-3 3-6 7-5.
There were also wins for qualifiers Juan Pablo Varillas and Nicolas Jarry against Lorenzo Sonego and Thiago Monteiro respectively.
Number seven seed Djere beat Spaniard Roberto Carballes Baena 7-5 7-6 (7-1) on the hard surface, while eighth seed Kwon comfortably saw off Japan's Yoshihito Nishioka 6-1 6-2.
Elsewhere, Thiago Monteiro earned an entertaining 6-2 3-6 7-6 (7-2) win over Daniel Altmaier, and the Brazilian will now face number three seed Marin Cilic in the round of 16 after the 2014 US Open champion was given a first round bye.
The other encounter on Monday saw Steve Johnson come through in three sets against wildcard Aleksandar Vukic, 6-4 2-6 6-4. The American will go up against second seed Karen Kachanov in the last 16.
Two-time Grand Slam semi-finalist Gael Monfils is the top seed for the ATP 250 tournament, with Cilic, Khachanov and Frances Tiafoe among the other big names who will enter the competition in the next round.
Slotted into the primetime fixture, Santiago resident Garin did not let the locals down, securing a break in the first game of the match to set the tone.
The other Chilean hopeful in the field, Nicolas Jarry, kept up his terrific form to also book his place in the second round.
After making the semi-final of last week's Rio Open – where he took a set off world number two Carlos Alcaraz – Jarry beat Peru's Juan Pablo Varillas 7-6 (7-1) 6-4 for an 11th victory from his past 14 matches.
Fifth seed Albert Ramos-Vinolas was the highest ranked player in action, and he went down 7-5 4-6 7-5 against Brazil's Thiago Monteiro, while sixth seed Laslo Djere had a better outing in his 6-3 6-2 victory over Joao Sousa.
Tomas Martin Etcheverry will meet second seed Francisco Cerundolo in an all-Argentine second round matchup after advancing past Italy's Fabio Fognini 6-1 7-6 (7-1).
Meanwhile, in Acapulco at the Mexican Open it was the Berrettini brothers stealing the headlines on a day where top hopes Carlos Alcaraz and Cameron Norrie both withdrew due to injuries.
World number 24 Matteo Berrettini won the first seven games of his match against Alex Molcan before the Slovak decided to retire at 6-0 1-0, and his younger brother Jacopo Berrettini capitalised on his qualifier status with a 3-6 7-6 (7-3) 2-1 (retired) comeback against Oscar Otte.
Australian eighth seed Alex De Minaur was the top seed in action and he discarded of Mexico's own 17-year-old qualifier Rodrigo Pacheco Mendez 6-1 6-2, while rising 21-year-old American Brandon Nakashima overcame Germany's Daniel Altmaier 6-3 3-6 6-1.
The biggest names in the field learned their opponents after the 32 seeded entrant received byes into the second round of the Masters 1000 event, and Tsitsipas will be desperate to avoid a third consecutive loss after quick exits at the Rotterdam Open and Indian Wells Open in recent weeks.
Fourth seed Daniil Medvedev is waiting for Roberto Carbellas Baena after he blew away Bernabe Zapata Miralles 6-0 3-0 (retired) in an all-Spanish showdown, while fifth seed Felix Auger-Aliassime will meet Brazil's Thiago Monteiro in the second round after his 6-4 7-6 (8-6) triumph over Australia's Jason Kubler.
After O'Connell and Kubler both fell, Thanasi Kokkinakis flew the flag for Australia with a 4-6 6-3 7-6 (9-7) comeback against Belgian wildcard Zizou Bergs, and his reward will be a battle against Polish eighth seed Hubert Hurkacz.
France's Gregoire Barrere earned a big matchup against 11th seed Cameron Norrie after eliminating Roman Safiullin 6-4 3-6 6-3, and Safiullin's Russian compatriot Pavel Kotov will join him on an early flight home after going down 6-2 6-2 against recent Chile Open finalist Tomas Martin Etcheverry.
Cordoba was the scene of Londero's breakthrough ATP Tour title last year, the Argentinian capping a fairytale week by overcoming countryman Guido Pella in three sets.
Back on the red clay following a first-round loss at the Australian Open in January, eighth seed Londero defeated Italy's Marco Cecchinato 6-2 7-5.
Londero – playing the night session – did not face a break point and won 70 per cent of his first serves, much to the delight of the home crowd at the ATP 250 tournament.
Next up for Londero is fellow Argentine player Pedro Cachin, who crushed Hugo Dellien 6-1 6-1 in just 58 minutes.
Spanish fifth seed Ramos-Vinolas beat local qualifier Facundo Bagnis 6-4 7-6 (7-3), despite wasting a match point at 6-5 in the second set.
It will be an all-Spanish affair in the last 16 after Pablo Andujar accounted for Filip Horansky 6-2 6-2.
Elsewhere, Jaume Munar (6-4 6-3 against Leonardo Mayer), Andrej Martin (7-6 [7-5] 6-4 over Federico Coria), Pedro Martinez (4-6 6-3 6-4 against Federico Gaio) and Corentin Moutet (6-4 6-3 over Thiago Monteiro) also progressed to the next round.
One of the top seeds in action, Giron is ranked over 300 spots higher than Rybakov, but the underdog put up a serious fight and held up well in all key indicators.
Rybakov had 29 winners with only 17 unforced errors, and he had just two double faults to go with seven aces, but Giron was better in both categories. His talent shone through with 34 winners and 16 unforced errors, while he also had 10 aces and no double faults.
Also in Dallas, French eighth seed Adrian Mannarino secured a 7-6 (8-6) 6-2 win against Steve Johnson, and he will now face Australia's Jordan Thompson following his 6-1 7-5 triumph over Denis Kudla.
Meanwhile, at the Cordoba Open, Argentina's Federico Coria was the top seed to advance after defeating Brazil's Thiago Monteiro 6-7 (5-7) 6-1 6-2.
There was less success for hometown seventh seed Pedro Martinez, who was eliminated 7-6 (7-3) 2-6 6-0 by Chile's Cristian Garin, while Federico Delbonis beat Alejandro Tabilo 6-7 (8-10) 6-2 7-6 (8-6) to earn a shot at second seed Francisco Cerundolo.
The highest-ranked Brazilian in the field made a winning start at South America's only ATP 500 event, but he had to work for it, emerging victorious after an 88-minute third set.
A raucous crowd cheered on every point down the stretch, and after saving four break point opportunities to force a tiebreaker, he jumped ahead to a 5-1 advantage and served it out.
He will face the winner between second seed Cameron Norrie and rising Argentine talent Juan Manuel Cerundolo in the next round.
Meanwhile, the top seed in action on Monday – third seed Lorenzo Musetti – continued his recent rough stretch of form with a disappointing 6-4 6-1 defeat at the hands of Chile's Nicolas Jarry.
Musetti, already ranked 18th in the world at 20 years old, has now lost four of his past five matches, while Jarry has won eight out of 10.
Jarry will play Spain's Pedro Martinez in the second round after he spoiled a potential all-Chilean matchup, eliminating Cristian Garin 7-6 (7-0) 6-4.
The only other Brazilian in action, Joao Fonseca, had no luck against Slovakia's Alex Molcan as he went down 6-0 6-3, while Bolivia's Hugo Dellien was similarly dominant in his 6-0 6-2 trouncing of Portugal's Joao Sousa.
After taking a set off world number two Carlos Alcaraz at last week's Argentina Open, Laslo Djere advanced into the second round as Facundo Bagnis retired in the second set while trailing 6-2 3-2.
Nishioka was the highest seed in action on the first day of the tournament, and he had his back up against the wall after failing to win a game in a troubling second set, but produced three breaks of serve in the back-and-forth decider.
Meanwhile, Ecuador's Emilio Gomez earned a shot at top seed and world number seven Taylor Fritz after advancing 7-6 (7-3) 6-4 against Taiwan's Tung-lin Wu.
Portugal's Nuno Borges continued his winning run after earning his spot through the qualifiers, eliminating America's Steve Johnson 6-4 3-6 6-4, but the crowd got something to cheer for when the USA's Denis Kudla got the better of Australia's Jordan Thompson 3-6 6-4 7-6 (7-2).
Further south at Argentina's Buenos Aires Open, Serbia's Laslo Djere booked a blockbuster showdown against world number two Carlos Alcaraz after repelling the challenge of Italy's Fabio Fognini 6-4 6-4.
Argentinian Pedro Cachin put on a show for his hometown fans with a 7-6 (7-2) 6-4 victory over Brazil's Thiago Monteiro, but Cachin's compatriot Guido Pella had less success with a 6-4 7-5 defeat at the hand of Spain's Jaume Munar.
Peniston beat world number five Casper Ruud as he reached the quarter-finals at the Queen's Club Championships last week, and followed that up by recovering from a set down against Rune to reach Eastbourne's last 16 in impressive fashion.
After wrapping up a 4-6 7-6 (7-5) 6-1 win, the 26-year-old told the home crowd: "I'm very happy with that. A tough start but I managed to fight, thanks to you guys.
"Since Queen's it has been madness. A couple of weeks ago was a lot different and things have changed, but I'm loving it."
Rune, who was twice two points from victory in an enthralling contest, was jeered by spectators after hitting a ball out of court and kicking his towel bin after being broken in the third set.
Peniston will face Pedro Martinez in the next round after he benefited from fellow Spaniard Alejandro Davidovich Fokina being forced to retire at one set apiece, while Ugo Humbert fell to a 6-7 (2-7) 6-3 6-4 reverse against Brazil's Thiago Monteiro.
Lorenzo Sonego posted a 4-6 7-6 (7-5) 7-6 (7-1) win over James Duckworth, while Tommy Paul recovered from a set down to beat Francisco Cerundolo and home favourite Dan Evans overcame Adrian Mannarino 6-4 6-3.
The seeds in action at the Mallorca Open endured mixed fortunes as Sebastian Baez cruised past Jordan Thompson in straight sets, but Botic van de Zandschulp was beaten by Marcos Giron.
The Dutchman succumbed to a 6-7 (6-8) 6-4 7-6 (7-2) defeat, while Germany's Daniel Altmaier beat Dusan Lajovic 7-5 7-6 (7-2).
Nick Kyrgios set up an enticing last-16 meeting with fifth seed Roberto Bautista Agut by knocking out Serbia's Laslo Djere in a marathon three-set contest, recovering to win 5-7 7-6 (7-1) 7-6 (7-1).
The 2022 US Open and French Open runner-up had struggled past Guido Andreozzi in the first round and followed up that up with a 7-5 2-6 7-6 (7-5) loss to Daniel.
The Japanese qualifier, ranked 125th in the world, won in just under three hours, winning the final three points of the deciding tie-break after Ruud led 5-4.
Ruud had more winners (38-32) but committed 30 unforced errors compared to Daniel's 12.
Taylor Fritz cruised into the last eight with a 6-4 6-4 win over Canada's Denis Shapovalov, setting up a quarter-final with sixth seed Francis Tiafoe, who eased past Feliciano Lopez 6-2 7-6 (8-6).
Fourth seed Holger Rune was too strong for Nuno Borges 6-0 6-2 inside an hour, while eighth seed Alex De Minaur crushed Jacopo Berrettini in just over an hour 6-1 6-0.
Top seed Lorenzo Musetti was a surprise second-round loser at the Chile Open with Jaume Munar triumphing 6-4 6-4 in one hour and 31 minutes.
Fourth seed Diego Schwartzman also bowed out, going down 6-4 4-6 7-6 (7-2) to local Nicolas Jarry.
Jarry will face Yannick Hanfmann in the quarter-finals, while Munar will take on Thiago Monteiro.
The Brazilian took just an hour and a half to see off the second seed 6-3 6-3 in Bastad on Wednesday.
Ruud successfully defended two break points in the opening game but could not gain the upper hand as Monteiro charged forwards, getting two breaks to take the set.
Monteiro continued to dominate in the second, though he only needed one break this time, and comfortably held off the Norwegian in the final game to make it into the next round, where he will face either Pavel Kotov or Duje Ajdukovic.
Data Debrief: Monteiro comes out on top
Monteiro did not give Ruud any sort of opening, winning all of his service games, and 75% of his service points (38/51).
The number two seed did not win more than one game in a row, and he will be hoping to put this result behind him quickly ahead of his doubles quarter-final with Rafael Nadal later on.
Team-mates Stefanos Tsitsipas and Despina Papamichail had already won their matches on Thursday, and Sakkari's 6-3 6-2 triumph gave Greece an unassailable 3-0 lead.
Michail Pervolarakis was defeated by Dimitar Kuzmanov in the final singles match between the nations, but Tsitsipas and Sakkari teamed up to earn Greece a 6-4 6-4 success in the doubles encounter to end on a high.
World number 16 Matteo Berrettini helped Italy topple Brazil by overcoming Thiago Monteiro 6-4 7-6 (9-7).
That gave Italy a 2-1 lead, and Lucia Bronzetti finished the job by demolishing Laura Pigossi in straight sets, with Brazil powerless to overturn the deficit despite Berrettini and Camilla Rosatello's defeat in the doubles match.
US Open semi-finalist Frances Tiafoe clinched a win for the United States over the Czech Republic after opponent Tomas Machac retired with an ankle injury with the score at 6-3 2-4.
Petra Kvitova had defeated Jessica Pegula 7-6 (8-6) 6-4 in the third singles match to give Czech Republic hope, but Tiafoe's win settled the contest before Pegula and Taylor Fritz joined up for success in the doubles.
Great Britain were 3-2 victors over Australia despite a pair of defeats on day two in Sydney.
Cameron Norrie's triumph over Nick Kyrgios' replacement Alex De Minaur on day one had helped Great Britain to a 2-0 lead over the hosts, and Harriet Dart prevailed over Maddison Inglis to wrap up the victory.
Dan Evans was then beaten 6-3 7-6 (7-3) by Jason Kubler before Dart and Jonny O'Mara fell to defeat in the doubles, but Great Britain's strong start saw them through.
France and Switzerland completed 5-0 sweeps of Argentina and Kazakhstan respectively, with Caroline Garcia winning both her singles and doubles matches in straight sets while Stan Wawrinka saw off Alexander Bublik 6-3 7-6 (7-3).
