Mujahed Salameh, the head of the Palestinian Petroleum Agency, said on Thursday that the deal was reached after Mahmoud Abbas sent a letter to Dor Alon guaranteeing to send funds within 10 days.
The fuel shortage had threatened to worsen economic problems that began when Western countries froze aid and cut most diplomatic contacts with the Hamas-led Palestinian Authority.
Dor Alon is the sole supplier of petrol and cooking gas to Palestinian areas.
Two days after it cut off deliveries on Tuesday, petrol stations ran dry across the occupied West Bank and traffic thinned.
"Dor will resume fuel supplies tomorrow [Friday] to the Palestinian areas," Salameh said, referring to the Gaza Strip and the occupied West Bank.
The Gaza Strip also felt the crunch, and petrol station owners in Gaza, home to 1.4 million Palestinians, said supplies were likely to be gone by the end of the day.
Abbas pledged to settle a portion
of the outstanding debt
In his letter to Dor Alon, Abbas pledged payment of 132 million shekels ($29.8 million), Salameh said.
He estimated the total debt to Dor Alon at 400 million shekels.
Salameh said the Palestian president agreed to take the money from the Palestine Investment Fund, the value of which has been estimated by Palestinian officials at close to $1 billion.
Abbas took control of it after Hamas won a parliamentary election in January.
Dor Alon had no immediate comment.
At one petrol station in Ramallah, the seat of Palestinian government, a sign read: "We ran out of fuel."
Salameh said earlier that factories, bakeries and public transport might have to stop work unless fuel supplies resumed.
Emergency service workers said they would be unable to reach patients in rural areas.