The tiny island nation, a self-governing part of the Netherlands with a population of little more than 150,000, are on the brink of becoming the smallest country to qualify for the World Cup if they avoid defeat in their last Group B qualifier in Kingston on Tuesday.
Otherwise, Jamaica, with former England manager Steve McClaren in charge, will finish top and advance to next year's finals in the United States, Canada and Mexico.
The 78-year-old Advocaat, who has coached seven other national teams in a long career, arrived with the team in Kingston at the weekend but then had to leave almost immediately for what the Curacao federation said were "family reasons".
A statement from Advocaat said: "It's a very difficult decision to have to leave the boys here. I had to make this decision with a heavy heart, but family is more important than football. From the Netherlands, I will stay in close contact with the staff and I have complete confidence in this group of players."
"It's bad news for us, but we understand that family is always a priority," goalkeeper Eloy Room, who is among 22 Dutch-born footballers in the Curacao squad, told Dutch NOS television.
"But it has changed nothing for us and our goal."
"We started this all together and now we want to finish it together," added captain Leandro Bacuna in an interview with Dutch daily Algemeen Dagblad on Monday.
"The coach doesn't have to worry, we will give everything against Jamaica," he added.
Curacao have a one-point lead at the top of the group standings after a 7-0 away win over Bermuda last Thursday, while Jamaica were held to a 1-1 draw in Trinidad and Tobago.
"We have come a long way. I remember being at an airport when no flight was arranged for us. Then I thought, 'How do we get home from a national team match?' But now everything is arranged properly and we have a strong selection," said Bacuna, who played in the 2015 FA Cup final with Aston Villa.
"This is a lot bigger. Then we were playing for a club, now it's for our country. You cannot compare it. We are going to do it here for the country and for our coach Dick Advocaat."