With its fuel supply line through Mozambique hit by sabotage, Zimbabwe today imposed stringent cutbacks on allocations to gasoline stations, touching off panic among motorists.

The Ministry of Industry and Energy Development, after a week of silence in the face of lengthening lines of cars at gas stations, announced new cuts in supplies and asked drivers to avoid unnecessary trips and panic buying.

The result this morning was lines of cars stretching a mile long in suburban areas and serpentine queues throughout the downtown area, often doubling up over four square blocks and snarling traffic.

The ministry did not announce the extent of the cutback, but garage owners said it was 40 percent, coming on top of a 10 to 20 percent cut in supplies last month.

Many African countries suffer fuel shortages, usually because of lack of foreign exchange to pay for the oil, but Zimbabwe's problem is different since it is a product of the struggle for black-majority rule in southern Africa.

Landlocked and without oil resources of its own, Zimbabwe must import all of its fuel via a pipeline from the Mozambican port of Beira or by rail from white-ruled South Africa. Zimbabwe's relations with the latter country are chilly.

Last week saboteurs blew up 34 tanks in Beira, leaving inadequate storage facilities for oil tankers bringing in fuel. Mozambique blamed the raid on South African commandos, but the Mozambique National Resistance Movement, which is seeking to overthrow the Marxist regime of President Samora Machel, claimed responsibility.

The pipeline from Beira to Mutare in eastern Zimbabwe has provided most of the country's fuel since reopening in June. It had been closed since 1965 when Rhodesia, the former name for Zimbabwe, declared independence to prevent black rule.

Since the pipeline reopened, it has been a target of the National Resistance Movement, which allegedly is backed by South Africa as part of its efforts to destabilize its black-ruled neighbors to maintain their dependence on Pretoria. South Africa routinely denies the charges.

The line has been damaged by explosives, a pumping station has been attacked and expatriate workers kidnaped. The violence has made it difficult for Mozambique to recruit foreigners to help operate the pipeline.

In recent months, Zimbabwe has based about 2,000 troops in Mozambique to help defend the vital oil line, and its troops are believed to have been involved in engagements with the rebels. This has threatened to carry the war across the border.

There are also reports that South Africa has slowed shipments of fuel by rail through the border town of Beitbridge. Although no confirmation of the reports was available, South Africa has in the past slowed down rail shipments to Zimbabwe at crisis periods.

Meanwhile, in scenes reminiscent of the 1979 fuel shortage in the United States as a result of the Iranian crisis, motorists in Harare are sometimes sleeping in their cars overnight at fuel stations to be first in line. Most stations are open for only several hours a day and usually sell less than four gallons per customer. Some allow only half that amount, for persons who often have waited two or three hours.

Offices are often unstaffed because employes are in fuel lines.

During the guerrilla war for independence, the Rhodesian government implemented a system of rationing that often limited drivers to about 1.5 gallons a week, but the government of Prime Minister Robert Mugabe has not made any moves to ration fuel. For conservation, the fuel contains a mixture of more than 20 percent ethanol.

The Herald newspaper, in which the government owns a controlling interest, said in a rare critical editorial today that the long silence by the administration had contributed to the air of panic.