New York's main contract, light sweet crude for January delivery, advanced 79 cents to 78.26 dollars a barrel in morning trade.
The December contract, which expired at the weekend, closed at 76.72 dollars in New York last Friday.
Brent North Sea crude for January delivery was up 69 cents to 77.89 dollars.
"We are seeing the US dollar weaken against the euro and investors continue to see oil at this low end of the trading range to be a buy opportunity," said Victor Shum, an analyst with energy consultancy Purvin and Gertz in Singapore. "Investors see the US dollar to be on a downward trend for the long term and so there are continued inflows into oil and other commodities."
Oil and other commodities such as gold are traded in US dollars and a weaker US currency makes them cheaper to holders of other units, driving demand and higher prices.
Shum said he expected oil to trade between the "high 70s and the low 80s" dollar-range until the end of the year.