The missile missed the drone and fell into the water, the official said.
Prior to taking fire, the American MQ-9 Reaper drone observed Iranian vessels closing in on the tankers, the official added, though the source did not say whether the unmanned aircraft saw the boats conducting an actual attack.
Still, it is the first claim that the US has information of Iranian movements prior to the attack.
CNN has not seen any imagery from the US drones.
Iran has strenuously denied any involvement in the attack.
The same official also said in the days prior to the attack, a US Reaper drone was shot down in the Red Sea by what is believed to be an Iranian missile fired by Houthi rebels.
The Pentagon had tough words for Iran on Friday as the US continues to assert its claim that Tehran was responsible for the attack on the two oil tankers in international waters after releasing video footage its says shows an Iranian patrol boat removing an unexploded mine from one of the vessels' hulls.
"We're making sure that General McKenzie and the central command has the resources and the support that they need to conduct their missions," acting Defense Secretary Patrick Shanahan said, adding that the US continues to work on building an international consensus that Iran was behind the attack.
The United Kingdom released a statement Friday saying it is "almost certain" that a branch of the Iranian military -- the Islamic Revolutionary Guard Corps (IRGC) -- attacked two tankers.
The IRGC is an elite wing of the Iranian military that was established in the aftermath of the country's revolution in 1979. In April 2019, the US officially designated it as a foreign terrorist organization, a move which was rejected by authorities in Tehran.
No other state or non-state actor "could plausibly have been responsible," the statement added.
Iran does not appear to be backing off and continues to engage in provocative behavior, according to a US official, who told CNN on Friday that Iranian small boats are preventing tug boats from towing away one of the damaged tankers.
But the owner of the Front Altair tanker has said they have "no information" and "no details" about any type of obstruction around its vessel since the time of the attack, Frontline spokesperson Pat Adamson told CNN Saturday.
President Donald Trump seems convinced that Tehran was responsible.
"Iran did do it and you know they did it because you saw the boat," he said, appearing to refer to the images and video released by the Pentagon late Thursday.
A senior diplomatic source of a US ally told CNN Friday, "It is a virtual certainty Iran was behind this latest attack. The video now nails it."
The source added that the goal of the response from the US government thus far is all about publicly exposing Iranian actions and intensifying maximum pressure.
After the United States withdrew from the Iran nuclear deal, it adopted a policy of maximum economic pressure and sent a carrier group to the region, the source said this "Iranian retaliation is designed to show they can disrupt, and push oil prices up."
The Trump administration predicted that the pressure from the United States would bring Iran to the negotiating table, but European leaders disagreed, believing it would empower the hardliners, the source continued.
Iran has denied any involvement in the incident, with its foreign minister suggesting that the US was quick to make allegations "without a shred of evidence."
The attack could provide more fodder for Iran hawks within the US administration, whose recent Iran saber-rattling has frustrated Trump. One of them, national security adviser John Bolton, announced last month that the Pentagon was deploying the USS Abraham Lincoln Carrier Strike Group and a bomber task force to the Middle East in response to a "number of troubling and escalatory indications and warnings" from Iran.
The two tankers -- one carrying oil and the other transporting chemicals -- were attacked near the strategically important Strait of Hormuz, a vital shipping route that has been the focal point of regional tensions for decades. Roughly 30% of the world's sea-borne crude oil passes through the strategic choke point, making it a flashpoint for political and economic friction.
Jonathan Cohen, acting US ambassador to the United Nations, said Thursday that he echoed Pompeo's comments in a private meeting of the UN Security Council, describing the attack as "another example of Iran's destabilizing activities in the region."
The Iranian mission to the UN rejected the US' claim.
Alireza Miryousefi, a spokesman for the Iranian mission, tweeted a statement saying Iran "categorically rejects the US unfounded claim" that Iran is behind the attacks and "condemns it in the strongest possible terms."
He added that Iran "expresses concern" over the "suspicious incidents." And he called it "ironic" that the US, which withdrew from the nuclear deal with Iran, the Joint Comprehensive Plan of Action (JCPOA), is now calling Iran to come back for negotiations and diplomacy.
Speaking on Friday at a conference in Tajikistan, Iranian President Hassan Rouhani said his country was willing to further scale back its commitment to the JCPOA if it does not receive "the necessary response from other signatories."
"We still believe that making good on commitments by all the signatories could have a major role in preserving regional and global peace and security. Iran will unilaterally not be able to remain as the party adhering to the JCPOA. Other parties need to do their own share as regards the deal," he said.
This story and headline have been updated.
CNN's Michelle Kosinski and Nada AlTaher contributed reporting