In the day's other tie, Switzerland scored a 2-1 victory over France after the duo of Celine Naef and Leandro Riedi beat Alize Cornet and Richard Gasquet 6-4, 7-5 in the mixed doubles rubber to book their place in Sunday's final.
Cornet earlier downed Naef 1-6, 6-3, 10-8 before Riedi got past Gasquet 3-6, 6-3, 10-8 to set up the decider.
Alcaraz, who won his second Grand Slam title on Sunday defeating 23-time major champion Novak Djokovic on the All England Club lawns, did not enjoy the smoothest shift back to clay in Nice, France, on his debut in the mixed team event.
Taking to the court after Elise Mertens beat Rebeka Masarova 7-6(3), 2-6, 10-5 to put Belgium 1-0 ahead, Alcaraz dropped his serve twice in sweltering conditions to surrender the opening set tamely to his seasoned opponent.
World number 111 Goffin, who defeated Alcaraz the last time they played after the youngster bagged the 2022 US Open crown, broke in the opening game of the next set and consolidated his lead to heap the pressure on his opponent.
Alcaraz responded by clawing back to 3-3 and rode his luck after both players wobbled on serve to level the match at a set apiece before switching on the after-burners from 0-4 down in the wildly swinging super tiebreak to wrap up the victory.
"It wasn't easy, it was a tough match but I'm really happy to come back to claycourts," Alcaraz said.
"It's a good surface for me and I really enjoyed playing tennis here.
"I felt the energy and the love from the people."
Alcaraz and Masarova, who replaced the injured Paula Badosa in the squad, were then thrashed 6-3, 6-1 by Goffin and Mertens in the mixed doubles rubber.
Spain will next take on Croatia on Saturday.
The relaunched Hopman Cup returned to the tennis calendar this week after a four-year absence, having lost its spot ahead of the Australian Open as a result of the short-lived ATP Cup men's team event.
The last edition of the Hopman Cup, which began in 1989 and was named after Australian great Harry Hopman, saw Roger Federer lead Switzerland to the title against Germany.
