Brent futures surged $12 in the first few minutes of business -- the most in dollar terms since they were launched in 1988 and representing a jump of nearly 20 percent -- while WTI jumped more than $8, or 15 percent.
Both contracts pared the gains but were both still more than 10 percent up.
The attack by Tehran-backed Huthi rebels in neighbouring Yemen, where a Saudi-led coalition is bogged down in a five-year war, hit two sites owned by state-run giant Aramco and effectively shut down six percent of the global oil supply.
Trump said Sunday the US was "locked and loaded" to respond to the attack, while Secretary of State Mike Pompeo said: "The United States will work with our partners and allies to ensure that energy markets remain well supplied and Iran is held accountable for its aggression."
Tehran denies the accusations but the news revived fears of a conflict in the tinderbox Middle East after a series of attacks on oil tankers earlier this year that were also blamed on Iran.
"Tensions in the Middle East are rising quickly, meaning this story will continue to reverberate this week even after the knee-jerk panic in oil markets this morning," said Jeffrey Halley, senior market analyst at OANDA.
Trump authorised the release of US supplies from its Strategic Petroleum Reserve, while Aramco said more than half of the five million barrels of production lost will be restored by tomorrow.
But the strikes raise concerns about the security of supplies from the world's biggest producer.
- Energy firms surge -
Oil prices had dropped last week after news that Trump had fired his anti-Iran hawkish national security adviser John Bolton, which was seen as paving the way for an easing of tensions in the region.
"One thing we can say with confidence is that if part of the reason for last week’s fall in oil and improvement in geopolitical risk sentiment was the news of John Bolton's sacking... and thoughts this was a precursor to some form of rapprochement between Trump and Iran, then it is no longer valid," said Ray Attrill at National Australia Bank.
The surge in crude lit a fire under energy firms, with Hong Kong-listed CNOOC up more than six percent and PetroChina five percent higher. Woodside Petroleum rallied more than five percent in Sydney.
However, airlines took a hit from the prospect of higher fuel costs. Cathay Pacific dropped almost three percent in Hong Kong Air China dropped two percent in Shanghai and Qantas dived more than four percent in Sydney.
Asian stock markets were mostly down after last week's rally, with investors awaiting a key Federal Reserve policy meeting this week hoping for another cut in interest rates.
Shanghai added 0.2 percent and Seoul gained 0.3 percent with Taipei 0.4 percent higher.
But Hong Kong sank more than one percent after fresh violent protests struck the city at the weekend, while Singapre shed 0.5 percent, Sydney gave up 0.1 percent and Jakarta sank 1.8 percent. There were also losses in Manila and Wellington. Tokyo was closed for a holiday.
On foreign exchanges, higher-yielding currencies dropped as traders shifted to safe-haven units such as the yen and dollar, while gold -- a go-to asset in times of uncertainty -- rose more than one percent.
The pound held its own around seven-week highs owing to easing fears of a no-deal Brexit.
- Key figures around 0230 GMT -
West Texas Intermediate: UP $5.66 at $60.51 per barrel
Brent North Sea crude: UP $6.93 at $67.15 per barrel
Hong Kong - Hang Seng: DOWN 1.1 percent at 27,047.99
Shanghai - Composite: UP 0.1 percent at 3,034.97
Tokyo - Nikkei 225: Closed for a public holiday
Pound/dollar: DOWN at $1.2481 from $1.2496 at 2030 GMT
Euro/pound: UP at 88.76 pence from 88.66 pence
Euro/dollar: UP at $1.1079 from $1.1078
Dollar/yen: DOWN at 107.74 yen from 108.11 yen
New York - Dow: UP 0.1 percent at 27,219.52 (close)
London - FTSE 100: UP 0.3 percent at 7,367.46 (close)
-- Bloomberg News contributed to this story --