Iran denies involvement in the attacks, which initially halved oil output from Saudi Arabia.
Iran will pursue any aggressor, even it carries out a limited attack, and seek to destroy it, the head of the elite Revolutionary Guards said on Saturday, after attacks on Saudi oil sites which Riyadh and U.S officials blamed on Tehran.
"Be careful, a limited aggression will not remain limited.
We will pursue any aggressor," the head of the Guards, Major General Hossein Salami, said in remarks broadcast on state TV.
"We are after punishment and we will continue until the full destruction of any aggressor."
Iran denies involvement in the attacks, which were claimed by Yemen's Houthi movement, a group aligned with Iran and currently fighting a Saudi-led alliance in Yemen's civil war.
Iran's foreign minister earlier denounced renewed US sanctions against its central bank following the Saudi attacks on Sept. 14 as an attempt to deny ordinary Iranians access to food and medicine, and said the move was a sign of US desperation.
The United States on Friday imposed a new round of sanctions on Iran, some aimed at its central bank and sovereign wealth fund.
"This is a sign of US desperation ... When they repeatedly sanction the same institution, this means their attempt at bringing the Iranian nation to its knees under 'maximum pressure' has failed," Foreign Minister Mohammad Javad Zarif told reporters in remarks shown on state television.
"But this is dangerous and unacceptable as an attempt at blocking ... the Iranian people's access to food and medicine," Zarif said, speaking after arriving in New York for the annual U.N. General Assembly next week.
Meanwhile, there were reports on social media that a number of Iranian servers and websites - including those of some petrochemical firms - were under a cyber-attack. There was no immediate official comment, and the websites of the main state oil company NIOC appeared to be functioning normally. Residents said their internet access was not affected.
NetBlocks, an organisation that monitors internet connectivity, said its data showed "intermittent disruptions" to some internet services in Iran starting on Friday evening.
But the group said the impact was limited, affecting only specific providers, and the cause was unclear. "Data are consistent with a cyber-attack or unplanned technical incident on affected networks as opposed to a purposeful withdrawal or shutdown incident," it said in a tweet.
NetBlocks Director Alp Toker said they saw four Iranian networks falling offline over a three hour period on Friday evening.
This began when the first reports emerged and ended shortly later, Toker said. The networks have been stable since.
He said the affected networks were all relatively small and only one is a consumer service provider, so there was no massive nationwide impact. But it is notable that these networks are usually stable, he said.
The fresh sanctions target the Central Bank of Iran, which was already under other US sanctions, the National Development Fund of Iran - the country's sovereign wealth fund - and an Iranian company that US officials say is used to conceal financial transfers for Iranian military purchases.
Zarif said he would on Wednesday meet foreign ministers of the remaining signatories to the 2015 nuclear accord, which was agreed with Britain, France, Germany, China and Russia as well as the United States.
"As we have said before, the United States can only attend if it returns to the (nuclear accord) ... and ends its economic war against Iran," Zarif said.
The United States withdrew from the accord last year and re-imposed unilateral sanctions on Iran.
"I hope the US government realises that they are no longer the only economic superpower in the world and that there are many countries that want to benefit from the Iranian market," Foreign Ministry spokesman Abbas Mousavi said, according to state media.