Early returns showed Rubio with 19,224 votes, followed by Donald Trump at 3,533 and Sen. Ted Cruz of Texas at 2,233 with a little more than half the island's voting centers reporting.
"The numbers are overwhelming," said local Rep. Jenniffer Gonzalez, chairwoman of Puerto Rico's Republican Party. "This primary in Puerto Rico ... will demonstrate that the Hispanic vote is important."
With Rubio holding 72 percent of the early vote, he seemed likely to finish with more than half and get all 20 delegates determined by the primary. If no one finished with more than 50 percent, the 20 delegates would be divided proportionately among candidates who received at least 20 percent. The territory's three super-delegates have committed to Rubio.
Residents of the U.S. island, which is struggling through a decade-long economic stagnation, cannot vote in the general election in November but can take part in the party primaries.
Orlando Gomez, a retiree whose two sons recently moved to the U.S. to seek jobs, said he participated in the primary because of concerns over the island's economy. He voted for Rubio.
"He understands Latinos because he is Latino," Gomez said. "And he can bring us closer to statehood."
Party officials estimated more than 30,000 people cast ballots Sunday. Officials said votes cast Friday by some 6,000 inmates would not be available until Wednesday.
Puerto Rico is one of the few U.S. jurisdictions that allow inmates to vote. The only other people allowed to submit absentee ballots in the territory were military members, but party officials said they did not yet know how many of those had been cast.
Any registered voter could take part, but only if they signed a document of affiliation with the Republican Party moments before casting a ballot.
Voters said in interviews that they want a U.S. president who will help Puerto Rico emerge from a decade of recession and a debt crisis that threatens basic government services.
"We need a radical change," Richard Suarez said as he waited to cast a ballot for Rubio in Guaynabo, a city in the San Juan metropolitan area. "Otherwise, we'll be stuck in the same situation and will have to keep asking the U.S. government for favors."
The island's economic woes affected the election itself. Edwin Mundo, electoral commissioner for Puerto Rico'sRepublican Party, said budget cuts forced it to reduce the number of polling places to 110 from more than 3,000 in 2012, dramatically reducing turnout.
No candidate has had a major presence in Puerto Rico this campaign, though Rubio visited the island Saturday for the second time in seven months. There have been no publicly released polls.
Jose Fuentes Agostini, Rubio's campaign president in Puerto Rico, said the senator's visits made a difference.
"This is a great victory," he said. "It is here in Puerto Rico that we're going to change the course of this presidential election for the entire nation."
Sonia Quintero, a retiree, praised Rubio for his visits and said she voted for him.
"He knows where we are and who we are and that's why he can help us," she said.
Politics in Puerto Rico generally revolves around the island's relationship to the U.S. One major party favors statehood and the other wants to maintain the current semi-autonomous commonwealth status. A small minority favors independence. The pro-statehood party has traditionally attracted Republicans and anyone taking part in the primary had to sign a pledge to support the effort to become the 51st state.