But West, the first-term incumbent, wasn't conceding and his campaign sought a hearing for further review of the votes, The West Palm Beach Post reported.
Murphy staked a claim to victory after the St. Lucie County Canvassing Board missed a noon deadline to finish its partial recount and submit its results to the Division of Elections. His lead had been widening as the recount progressed.
WPTV-TV, West Palm Beach, reported that after more than 37,000 St. Lucie County ballots were reviewed Canvassing Board member Tod Mowery said Murphy had 65,841 votes to 52,704 for West.
While the results have to be certified by the governor and secretary of state, Murphy campaign senior adviser Eric Johnson said the outcome was clearly in Murphy's favor.
"As far as we're concerned, this puts an end to it," Johnson said. "The West people demanded a recount. They got that retabulation and Patrick Murphy gained a margin of 242 votes.
"This election is over."
Murphy's unofficial lead is greater than the 0.5 percent difference needed to trigger an automatic recount under state law.
West's campaign manager, Tim Edson, said the situation was under review.
"We are still evaluating how to handle this," he said. "It's a unique case."
The Post reported Edson said Gov. Rick Scott and Secretary of State Ken Detzner "will have to take a look at these recounts."
The Post said as soon as the noon deadline passed, attorneys for West went to the St. Lucie County Courthouse to request an emergency hearing on the issue. It was not known if one was scheduled, the newspaper said.
Florida law allows an exemption to the deadline in an emergency and West's attorney are expected to argue that the exemption applies.