Michael Hosam, aged 66, received medical attention at the ground before he was taken to the Port-of-Spain General Hospital where he was pronounced dead.
"He was a true West Indies fan and he loved the team. He loved West Indies cricket," his brother-in-law Ikins Williams said.
Other fans attending the game had noticed Hosam looking unwell before informing medical personnel at the ground.
"I would like to extend my condolences to Mr. Hosam's family and friends on this sad occasion," Julian Hunte, president of the West Indies Cricket Board said. "I hope everyone close to him will be able to cope with this loss.
"We in West Indies Cricket know we have people out there who are loyal and dedicated in their support and we wish his family all the best at this time."
West Indies, replying to the visitors' first innings total of 278, reached 268 for seven when bad light stopped play 14.1 overs early.