The cause of the 53-year-old's death wasn't immediately clear.
He was performing at Hamden Town Center Park when he collapsed on Friday evening, town chief of staff Sean Grace said on Saturday.
Elliott led celebrity tributes to the singer, saying his voice had made people want to dance for more than 20 years.
"Prayers for Fatman Scoop family for STRENGTH during this difficult time," Elliot posted on X.
Elliott praised Scoop's "VOICE and energy", saying he had contributed to many songs that made people happy over more than two decades.
"Your IMPACT is HUGE & will NEVER be forgotten," she added.
Elliott worked with Scoop on her hit 2005 single Lose Control.
Hamden Mayor Lauren Garrett posted on Facebook the rapper had a medical emergency. Concertgoers and paramedics tried to help the artist, who was taken to a hospital, she said.
His family said in an Instagram post that "the world lost a radiant soul, a beacon on stage and in life".
With a gravelly voice and dance-floor-friendly sensibility, Fatman Scoop was a mainstay of club playlists around the turn of the millennium.
But if the world knew him as the "voice of the club", his family cherished him as "the laughter in our lives, a constant source of support, unwavering strength and courage", his relatives said.
"His music made us dance and embrace life with positivity. His joy was infectious and the generosity he extended to all will be deeply missed but never forgotten," they added, saying he leaves a legacy "of love and brightness".
Born Isaac Freeman III, Fatman Scoop was from New York City's Harlem neighbourhood and broke out with 1999's Be Faithful.