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.
Korda was playing his first indoor match since the decider of the Next Gen ATP Finals last November, and Hugo Gaston represented a tough opening test.
But the American prevailed in straight sets against the home favourite, winning 6-3 7-6 (7-2) after defending set points in the second set and blowing his opponent away in the subsequent tie-break.
Another tricky opponent awaits, with number three seed Lorenzo Musetti up next in the second round.
Musetti's Italian compatriot Sonego also succeeded in Monday's only other main-draw match, beating sixth seed Aslan Karatsev 7-5 6-3 to set up a last-16 clash with either Gilles Simon or David Goffin.
In the qualifying section, three-time grand slam champion Wawrinka beat Zsombor Piros 6-7 (4-7) 6-2 6-0 to reach the first round, where Joao Sousa awaits.
The winner of that match will face top seed Daniil Medvedev in the last 16.
Sixth seed Sonego was the headline act on Monday in Stuttgart and made light work of Benoit Paire to win 7-5 6-2.
The Italian will next face Jan-Lennard Struff after the German edged out world number 53 Marcos Giron 7-5 5-7 7-6 (10-8).
"It's my first time here, it's my first tournament on grass and I'm really happy [with] this match," Sonego said after defeating Paire.
"It's never easy to pass from clay to grass, but I enjoy it because if I serve good it is easier for me."
Struff's compatriot Oscar Otte was another first-round winner with straight-sets victory over Daniel Altmaier, while Frenchman Benjamin Bonzi breezed past Feliciano Lopez.
In Rosmalen, poor weather wreaked havoc with the schedule but France's Gaston had little trouble against South Korean Kwon Soon-woo, winning 6-2 6-1 to reach the last 16, and Andreas Seppi battled past Gijs Brouwer in the second round of qualifying.
Seppi will be hoping to make it to the main draw, where world number two Daniil Medvedev is the top seed, with Felix Auger-Aliassime and Taylor Fritz also competing.
