Brent crude futures climbed 79 cents, or 1.26%, to $63.26 a barrel by 1225 GMT.
West Texas Intermediate (WTI) crude futures were up 74 cents, or 1.33%, at $56.37 a barrel.
Last week, WTI fell over 7% and Brent lost more than 6%.
"The events in the Gulf have definitely taken the market into more bullish territory in today's trading," said Erik Norland, senior economist at CME Group.
"But that doesn't mean markets will continue to go higher, and previous incidents in the Gulf haven't driven up prices much - suggesting that investors' calculus, rightly or wrongly, is that a war is not very likely."
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 the Gulf, through which flows about one-fifth of the world's oil supplies, but no major escalation with Britain or the United States appears imminent.
"In the cat and mouse game that Iran is playing with the U.S., it is taking calculated risks," Harry Tchilinguirian, global oil strategist at BNP Paribas in London, told the Reuters Global Oil Forum.
"So far the U.S. is not taking the bait."
Capping gains, force majeure was lifted on loadings of crude on Monday at Libya's Sharara oilfield, the country's largest, whose closure since Friday had caused an output loss of about 290,000 barrels per day (bpd).
Meanwhile, data late last week showed shipments of crude 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 oil in response to the escalating dispute between Iran, the United States and other Western nations, along with signs of falling supply.
The Iranian capture of the ship in the global oil trade's most important waterway was the latest escalation in three months of confrontation with the West that began when new, tighter U.S. sanctions on Iran took effect at the start of May.
Hedge funds and other money managers raised their combined futures and options 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 forecast of growth in oil demand for 2019 to 1.275 million bpd, citing disappointing global economic activity.