Futures extended gains after weekly government data showed US crude stockpiles fell by 1.1 million barrels.
Since the start of the supply cuts, crude inventories have gradually declined from record levels towards long-term average levels.
"I do think we could see $70 pretty quick, but I want to caution that maybe we'll see the market level out a little bit in a few weeks", said Phil Flynn, analyst at Price Futures Group in Chicago. US shale producers are likely to keep pumping at record levels as prices rise to exploit spreads between West Texas Intermediate and Brent prices.
Japan's Nikkei faded late in the day to end up 0.15 percent, but basic materials and utilities both climbed more than 2 percent.
The EIA reported USA crude stockpiles dropped 1.07 million barrels last week, while supplies at the Cushing, Oklahoma, pipeline hub dropped by 1.12 million.
Stockpiles of gasoline also dropped by 3 million barrels, while distillates fuels including diesel decline by 3.1 million barrels.
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"It's a bullish report with the across-the-board drawdowns in everything", John Kilduff, founding partner at energy hedge fund Again Capital, told CNBC.
"Overall the supply-demand equation is fairly balanced", said Anthony Scott, managing director at BTU Analytics in Denver.
Opec, which has been curbing output by 1.8 million barrels a day since January 2017, will need to "over-tighten the market rather than under-tighten the market", to keep United States shale from flooding the sector, said Michael Tran, managing director, global energy strategy at RBC Capital Markets in an interview with The National in Abu Dhabi.
Yields on USA two-year Treasuries stood at levels last visited in 2008 at 2.43 percent. The total global supply controlled by OPEC, with Saudi Arabia as its de facto leader, has dropped to 40% in recent years, mainly due to the expansion of USA drilling. Saudi Arabia ran budget deficits that were unthinkable before 2014, and its breakeven budget price is said to be upwards of $80 oil. The current instability and war situation in the Middle East Region is adversely affecting the oil prices. The cartel has used its power to manipulate market prices for oil in the past and actually caused their own present heartache when they dumped cheap oil on the global market trying to drive out competition from U.S. frackers, among other reasons.