Price gains were, however, capped to some degree by an unusually downbeat OPEC report that stoked concerns about growth in oil demand.
Brent crude, the international benchmark for oil prices, was up 36 cents, or about 0.6pc, at $59.00 a barrel at 1225 GMT.
US West Texas Intermediate (WTI) crude futures were up 30 cents, or 0.5pc, at $55.17 a barrel.
A drone attack by Yemen's Houthi group on an oilfield in eastern Saudi Arabia on Saturday caused a fire at a gas plant, adding to Middle East tensions, but state-run Saudi Aramco said oil production was not affected.
"The oil market seems to be pricing in again a geopolitical risk premium following the weekend drone attacks on Saudi Arabia, but the premium might not sustain if it does not result in any supply disruptions," said Giovanni Staunovo, oil analyst for UBS.
Iran-related tensions appeared to ease after Gibraltar released an Iranian tanker it seized in July, though Tehran warned the United States against any new attempt to seize the tanker in open seas.
Concerns about a recession also limited crude price gains.
Meanwhile, China's announcement of key interest rate reforms over the weekend has fuelled expectations of an imminent reduction in corporate borrowing costs in the struggling economy, boosting share prices on Monday.
US energy firms this week increased the number of oil rigs operating for the first time in seven weeks despite plans by most producers to cut spending on new drilling this year.
"WTI in recent weeks has performed relatively better than Brent... Pipeline start ups in the United States have been supportive for WTI, while the ongoing trade war has had more of an impact on Brent," said Warren Patterson, head of commodities strategy at Dutch bank ING.
The Organization of the Petroleum Exporting Countries (OPEC) cut its forecast for global oil demand growth in 2019 by 40,000 barrels per day (bpd) to 1.10 million bpd and indicated the market would be in slight surplus in 2020.
It is rare for OPEC to give a bearish forward view on the market outlook.
"Such a bearish prognosis will heap more pressure on OPEC to take further measures to support the market," said Stephen Brennock of oil broker PVM.