Tillerson's first official visit to the United Kingdom comes after senior UK government officials lambasted the United States
over the leaks and temporarily suspended intelligence sharing on the investigation.
But Tillerson, addressing reporters alongside Johnson after signing a book of condolence for victims of the Manchester attack, said the spat would not impact on the two countries' longstanding friendship.
US President Donald Trump "has been very strong in his condemnation," Tillerson said, and has called for the investigation and prosecution of those responsible for leaking the information.
"We take full responsibility for that and we obviously regret that that happened," he said. "In terms of how to fix the relationship between the United States and Great Britain, the special relationship that exists between our two countries will certainly withstand this particular unfortunate event."
Tillerson also voiced his sorrow for the Manchester victims, saying that "all across America hearts are broken, broken at the very thought of the loss of life of loved ones, the injured, the effect it has on their families."
The suicide bombing claimed the lives of 22 people, many of them children, and injured dozens more.
US sources were the first to reveal the identity of the suicide bomber, Salman Abedi, leading to concern that police efforts to hunt down his associates could be impacted. The leaks culminated in the New York Times publishing crime scene photos
Tillerson: 'Drive out extremists'
Tillerson vowed to stand alongside the British people in the fight against terrorism, saying ISIS' decision to target a "concert full of children" was reprehensible.
"Even as our ally and friend mourns, the fires for justice burn very hot in all of our hearts," he said.
"We will drive out the terrorists and extremists. As President Trump said earlier this week in Saudi Arabia, we must drive the extremists out of our communities, we must drive them out of any country that would provide them with safe haven and we must drive them off the face of the earth."
Johnson, who described Tillerson's visit as an "instinctive act of solidarity between the US and the UK," said the two nations would stand together around the world in defense of democracy and the rule of law.
He said the pair had also discussed Syria, relations with Iran and North Korea, and "the vital importance of the work we do together across such a range of fields, including, of course, intelligence sharing."
"Deeply troubling' leaks
The US State Department said Tillerson's visit was intended to reaffirm Washington's commitment to the "special relationship between the United States and the United Kingdom and our solidarity in defeating terrorism in every part of the world."
UK Prime Minister Theresa May confronted Trump about the intelligence leaks during their meeting at the NATO headquarters in Brussels, Belgium, on Thursday.
The breakdown of trust between the two countries led to the brief suspension on the sharing of intelligence on Thursday. Later that day, after receiving "fresh assurances," the suspension was lifted, according to the National Police Chiefs' Council.
White House leaks nothing new
The Manchester intelligence leaks are not the first to have emanated from the White House since Trump took office.
His administration has been beset by whistleblowers
since January, much to the President's frustration, and he has repeatedly called for harsher penalties for those who leak sensitive information to the press.
Trump himself was widely criticized for allegedly sharing highly confidential information
with the Russian Foreign Minister and the Russian Ambassador to the US in an Oval Office meeting.
H.R. McMaster, Trump's national security adviser, denied that sensitive intelligence was divulged.
"At no time were intelligence sources or methods discussed and the President did not disclose any military operations that weren't already publicly known," he said. "I was in the room. It didn't happen."