Expectations that the US Federal Reserve could cut interest rates at its next meeting, stimulating growth in the world's largest oil-consuming country, and a drop in US crude inventories, also provided support to prices.
Brent crude, the global benchmark, was up $1.40 at $63.22 a barrel at 1035 GMT, having earlier risen 3.3% to $63.88. US West Texas Intermediate crude rose $1.54 to $55.30.
"The risk of a military conflict in the Middle East has risen because of a ratcheting up of tensions between the United States and Iran," said Abhishek Kumar of Interfax Energy in London.
"Elsewhere, the US Federal Reserve has signaled its willingness to loosen monetary policy over the coming months, which is being perceived as favourable to oil demand."
The drone was downed in international airspace over the Strait of Hormuz by an Iranian surface-to-air missile, a US official said. Iran's Revolutionary Guards said the drone was flying over southern Iran.
Tension has been rising in the Middle East, home to over 20% of the world's oil output, after attacks on two tankers near the Strait of Hormuz, a chokepoint for oil supplies. Washington blamed Tehran for the tanker attacks. Iran denied any role.
Concern about slowing economic growth and a US-China trade dispute has pulled oil lower in recent weeks. Brent reached a 2019 high of $75 in April.
The prospect of further rate cuts could prove the more significant factor for oil, said Petromatrix analyst Olivier Jakob, should Iran-US tension not escalate. "The Fed and the cutting of rates is something that will provide more substantial support," he said.
Also propelling oil higher on Thursday was a decline in US crude inventories and the prospect of prolonged supply restraint by Organisation of the Petroleum Exporting Countries (OPEC) and its allies.
US crude stocks fell by 3.1 million barrels last week, more than analysts expected, the Energy Information Administration said on Wednesday.
OPEC and allies including Russia agreed this week to meet on July 1-2, ending a month of wrangling about the timing.
The coalition known as OPEC+ looks set to extend a deal on cutting 1.2 million barrels per day of production. The deal expires at the end of June.
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