Brent crude futures climbed 88 cents, or 1.4%, to $63.35 a barrel by 0706 GMT.
West Texas Intermediate (WTI) crude futures were up 58 cents, or 1%, at $56.21 a barrel.
WTI fell over 7% and Brent fell more than 6% last week.
"Falling global demand and rising U.S. stockpiles have helped turn oil charts very bearish, but that may not last as tensions remain high in the Persian Gulf," Edward Moya, senior market analyst at OANDA in New York, said in a note.
Iran's Revolutionary Guards said on Friday they had captured a British-flagged oil tanker in the Gulf in response to Britain's seizure of an Iranian tanker earlier this month.
The move has increased the fear of potential supply disruptions in the Strait of Hormuz at the mouth of Gulf, through which flows about one-fifth of the world's oil supplies.
Britain was weighing its next moves on Sunday, with few good options apparent as a recording emerged showing that the Iranian military defied a British warship when it boarded and seized the ship.
Meanwhile, a senior United States administration official said on Friday the U.S. will destroy any Iranian drones that fly too close to its ships, a day after the U.S. said one of its navy ships had "destroyed" an Iranian drone in the Strait of Hormuz. Iran said it had no information about losing a drone.
Crude oil supply outages and curbs also helped lift prices higher.
"Oil prices got a small boost this morning after Libya's (NOC) declared force majeure on Sharara crude loaded at Zawiya port," said Stephen Innes, managing partner at Vanguard Markets.
The Sharara oilfield resumed production at half capacity on Monday after being shut down since Friday, which caused an output loss of about 290,000 barrels per day (bpd).
Meanwhile, data late last week showed shipments of crude oil from Saudi Arabia, the world's top oil exporter, fell to a 1-1/2 year low in May.
Speculative money is flowing back into the oil markets in response to the escalating dispute between Iran and the United States and other western nations playing out in the Gulf waters along with the signs of falling supply.
Hedge funds and other money managers raised their combined futures and option's positions on U.S. crude for a second week and increased their positions in Brent crude as well, according to data from the U.S. Commodity Futures Trading Commission and the Intercontinental Exchange.
Goldman Sachs on Sunday lowered its year-on-year oil demand forecast for 2019 to 1.275 million bpd, citing disappointing global economic activity.
The forecast was still above the consensus of about 1.05 million bpd for 2019, it said, adding that "we see increasing scope for oil demand to finally start exceeding beaten-down expectations."