Despite the move, spare capacity from other suppliers such as Saudi Arabia might be able to ensure oil markets cope with a cut in Iranian exports.
Brent crude futures were at $74.67 per barrel at 0855 GMT, up 63 cents or 0.85 percent from their last close, after hitting their highest level since November at $74.70.
U.S. West Texas Intermediate crude futures marked their strongest since October 2018 at $66.14 per barrel, up 59 cents or 0.9 percent from their previous settlement.
The United States on Monday demanded that buyers of Iranian oil stop purchases by May 1 or face sanctions, ending six months of waivers which allowed Iran's eight biggest buyers, most of them in Asia, to continue importing limited volumes.
Before the reimposition of sanctions last year, Iran was the fourth-largest producer among the Organization of the Petroleum Exporting Countries at around 3 million barrels per day (bpd), but April exports have shrunk to below 1 million bpd, according to tanker data and industry sources.
U.S. President Donald Trump is confident that Saudi Arabia and the United Arab Emirates will fulfill their pledges to make up the difference in oil markets, a U.S. official told reporters.
Saudi Energy Minister Khalid al-Falih said on Monday that his country would "coordinate with fellow oil producers to ensure adequate supplies are available to consumers while ensuring the global oil market does not go out of balance."
Saudi Arabia is the world's top oil exporter and de facto leader of OPEC, which has led global supply cuts since the start of the year aimed at propping up crude prices.
The group is set to meet in June to discuss output policy.
Barclays bank said in a note that the U.S. decision took many market participants by surprise and would "lead to a significant tightening of oil markets."
The move to increase pressure on Iran came amid other sanctions Washington has placed on Venezuela's oil exports.
U.S. bank Goldman Sachs said an expected decline of 900,000 bpd in Iranian exports stood "versus immediately available and demonstrated spare capacity of 2.0 million bpd, which is set to grow further later this year."