Brent futures rose $1.63, or 2.3pc, to settle at $72.70 a barrel, while U.S. West Texas Intermediate (WTI) crude rose $1.32, or 2.0pc, to settle at $68.74
Brent crude oil futures were up 92 cents, or 1.2pc, at $75.66 a barrel by 12:05 ET (1605 GMT), having traded as high as $75.74. U.S. West Texas Intermediate (WTI) crude oil futures were up 90 cents, or 1.3pc, at $73.29 a barrel.
Brent crude rose 87 cents, or 1.2%, to $75.68 by 12:06 p.m. ET (1606 GMT). Its session high, $76.02 after the EIA data, was the highest since October 2018.
The softer dollar, down 0.1pc, reduced gold's cost for holders of other currencies. Markets broadly shrugged off U.S. PMI data showing factory activity at a record high in June.
"Oil markets are rallying because OPEC is sceptical that the increase in U.S. oil production is going to be enough to change their plans to support prices," said Phil Flynn, senior analyst at Price Futures Group in Chicago.