The move is the latest in months of wrangling over the controversial policy, which was approved by the government at the height of the country's Covid-19 outbreak but then blocked by Israel's top court pending new legislation.
Cabinet mandated Shin Bet to use cell phone surveillance as an emergency measure to combat the virus in mid-March as mounting numbers of Israelis tested positive for the novel coronavirus.
The specifics were kept secret, but security officials said the agency had tracked virus carriers' movements through their phones.
The measure was then challenged in Israel's top court, which ruled that it must be formalised in law or dropped.
The government chose to not to push forward with legislation and the measure was discontinued on June 10 as infection rates dropped.
But following two weeks that have seen growing numbers of Israelis infected with the virus, Prime Minister Benjamin Netanyahu decided to table the bill.
"Contagion has already crossed the 400-person (daily) threshold, and unfortunately indicates a continued rise" in novel coronavirus cases, he said Wednesday ahead of a cabinet vote on the move.
He said a civilian application was needed that would inform people they had been near coronavirus carriers, adding that until an alternative was ready, Shin Bet would be needed.
"I hope we won't even have to use" the agency, he said.
The bill passed its first plenary reading 45-32, ahead of a debate in a parliamentary committee.
A second and third vote will then be needed in the 120-seat legislature, expected next week.
Israel, which has a population of around nine million, announced its first coronavirus patient on February 21.
Since then it has confirmed more than 21,730 cases, including 308 deaths.
Tehilla Shwartz Altshuler from the Israel Democracy Institute think-tank said the Shin Bet's cellphone tracking abilities were limited and flawed, adding that the use of the spy agency provided a "false sense of security" and was a "privacy violation".
Hostility to the policy is not limited to opposition political parties and NGOs; Shin Bet chief Nadav Argaman has consistently urged cabinet to find an alternative.
In a recording leaked to Channel 12 television, he told ministers on Sunday he was against legislation binding his organisation to virus tracking at this stage.
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