The match in Cagliari, one of 12 qualifiers to decide which nations go to the Madrid finals in November, is being held in an empty stadium because of the coronavirus crisis that has hit Italy especially hard.
Fognini beat Lee Duck-hee 6-0 6-3 before Gianluca Mager, making his debut, beat Nam Ji-Sung 6-3 7-5 - both matches played in front of team officials and some media.
"Conditions were strange," Fognini told the Davis Cup website. "I guess I am the first in Italy to play a Davis Cup match in these conditions.
"It was not difficult to focus, but it was a bit strange to play in front of 100 people. But safety first and then sport."
Japan were also playing their tie against Ecuador in an empty stadium in Miki because of the coronavirus spread, which has played havoc with sporting fixtures across the world.
Without Kei Nishikori, Japan trailed 2-0 after Roberto Quiroz beat Yasutaka Uchiyama and Emilio Gomez beat Go Soeda.
Elsewhere 2018 champions Croatia are close to booking their spot in Madrid after establishing a 2-0 lead over India in Zagreb. Marin Cilic edged past Ramkumar Ramanathan 7-6 7-6 after Borna Gojo beat Prajnesh Gunneswaran in three sets.
In a local derby Czech Republic lead Slovakia 2-0 in Bratislava, while Germany and Belarus were level 1-1 in Dusseldorf.
Pablo Cuevas beat Jurij Rodionov to take Uruguay into the second day against a Dominic Thiem-less Austria level at 1-1. Uruguay have never been in the World Group before.
The Netherlands were locked at 1-1 away in Kazakhstan, as were Sweden and Chile in Stockholm and Hungary and Belgium in Debrecen.
Australia will continue on Saturday with a healthy 2-0 lead over Brazil. The other two ties feature United States at home to Uzbekistan and Colombia hosting Argentina.