"There will be train service operating all day," federal mediator George Cohen said late Tuesday night.
Both sides had made progress in their talks, he said, declining to offer details.
"Talks continue," the rapid transit system said in a Twitter message late Tuesday night after saying mediators agreed there would be no strike.
The talks were expected to last into the wee hours Wednesday, after an 11:59 p.m. Tuesday strike deadline.
"We're working very hard to achieve an agreement," said Josie Mooney, chief negotiator for Service Employees International Union 1021, representing 2,300 line-level employees.
She was quoted by the San Francisco Chronicle as saying she hoped both sides would stay at the bargaining table "until they got a deal."
Mooney said Cohen, Washington's top government mediator, offered some creative solutions.
"A lot of progress has been made, but there still is a lot to be worked out," BART spokeswoman Alicia Trost said Tuesday evening outside the California Department of Transportation office in Oakland, where the talks were held.
A gag order was in place, but the San Jose Mercury News said BART was considering a union counterproposal after submitting what it said was its final offer Sunday.
BART drivers and other employees have worked without a contract since June 30.
They employees walked off the job for 4 1/2 days in July. Two other strikes were averted when Democratic Gov. Jerry Brown stepped in and sought a 60-day cooling-off period.
The cooling-off period expired at 11:59 p.m. Sunday, hours after BART offered the "final offer."
The unions agreed that night to talk one more day, then extended the deadline to 11:59 p.m. Tuesday after reporting progress.
The San Francisco Bay Area Rapid Transit District, which operates the fifth-busiest U.S. heavy-rail rapid-transit system, has said it wants workers to contribute to pensions and wants a cap on its healthcare costs.
Meanwhile, the Alameda-Contra Costa Transit District, known as AC Transit, said Tuesday it asked Brown for a 60-day cooling-off period to delay a bus strike threatened in the East Bay for Thursday.
The cooling-off period would force workers to stay on the job as the two sides continued to negotiate.
The Oakland bus system's board of directors requested the period Oct 8 but only disclosed it Tuesday, a day after union workers formally issued a 72-hour strike notice.
Brown's office said late Tuesday it was reviewing the request.