A massive column of black smoke billowed from the area around the port on Tuesday, a Reuters video showed, and a witness said blasts had been heard from other areas though it was not clear exactly where else had been hit.
Sudan's electricity company said a substation in the city was also hit, causing a complete power outage, part of a systematic assault on infrastructure.
Port Sudan had enjoyed relative calm since the civil war erupted in April 2023, becoming the base for the army-aligned government after the Sudanese armed forces lost control of much of the capital Khartoum at the start of the conflict to the paramilitary Rapid Support Forces.
Hundreds of thousands of displaced people have also sought refuge in the city, where UN officials, diplomats and agencies have also set up headquarters, making it the main base for aid operations in what the UN has called the world's biggest humanitarian disaster.
Port Sudan's import and storage depots supply fuel across the country and the destruction of its facilities risks a major crisis, throttling aid deliveries by road and hitting electricity production and cooking gas supplies.
The attacks, which began on Sunday, open a new front in the conflict, targeting the army's main stronghold in eastern Sudan after it drove the RSF back westwards across much of central Sudan, including Khartoum, in March.
Military sources have blamed the paramilitary RSF for the attacks on Port Sudan since Sunday.
On Sunday drones struck a military base in the area near Sudan's only functioning international airport, and on Monday they targeted the city's fuel depots.
One of the targets was a major hotel near the residence of Sudan's military leader, General Abdel Fattah al-Burhan, witnesses said.
The attacks came after a military source said the army had destroyed an aircraft and weapons depots in the RSF-controlled Nyala airport in Darfur, the main stronghold of the paramilitary group.
The attacks have drawn condemnation from neighbouring Egypt and Saudi Arabia, as well as concern from the United Nations.
Sudan's army-aligned government has accused the United Arab Emirates of backing the RSF, accusations that UN experts have found credible.
The UAE denies backing the RSF and the International Court of Justice on Monday said it could not rule in a case in which the government accused the UAE of fuelling genocide.
The war, triggered by a dispute over a transition to civilian rule, has displaced more than 12 million people and pushed half the population into acute hunger, according to the United Nations.
With the army's success in pushing the RSF out of most of central Sudan, the paramilitary has shifted tactics from ground incursions to drone attacks deep in army-controlled territory.
