Mr Tillerson also met UK Prime Minister Theresa May, and the Iran nuclear deal - which has been heavily criticised by Mr Trump - was on the agenda.
"Sometimes we forget about the importance of our own relationship", he said.
"We are concerned about the Turkish incident in Northern Syria", Tillerson said while on a trip to London, according to a pool report supplied to Reuters.
'It is always worth saying that the relationship between the United States and the United Kingdom is absolutely fundamental to our diplomacy but also to our economy, ' he said.
The pair talked about the Iranian nuclear agreement, Yemen and Turkey's assault on Kurdish militia in Syria.
"We will also have the opportunity of underlining our firmness on Iran's compliance with United Nations Resolution 2231, which limits access to ballistic capacity and which Iran does not respect", Le Drian said Monday in Brussels ahead of the meeting with his European Union counterparts.
Downing Street said in a statement the pair agreed on the need for the global community to unite in countering Iran's "destabilising regional activity", while the prime minister also reiterated the Britain's commitment to the nuclear agreement.
The U.S. sold its old embassy building in 2009, and developers broke ground on the new site in 2013.
The American flag was this month removed from Grosvenor Square where the US embassy has been based since 1938 with the area known as "Little America" during World War Two, when the square also housed the military headquarters of General Dwight D. Eisenhower.
Trump said this month that he canceled a planned trip to London because he didn't want to cut the ribbon at the high-tech new building, thought to be the most expensive embassy in the world.More news: Bay Area serial stowaway arrested at Chicago airport
"I think there's a common view among the E3 ( France, Germany and the UK) that there are some areas of the [nuclear deal] or some areas of Iran's behavior that should be addressed", Tillerson told reporters after talks with Johnson.
But his decision came after a series of rows and after it became clear the visit would be met with mass protests.
Nauert said the new embassy opened this week and the bulk of staff had moved from the old building.
The embassy opened its doors to the public last week.
Woody Johnson, the US ambassador to Britain, greeted Tillerson at the $1.2 billion building as workmen completed planting shrubs on the grounds, Reuters reported.
Asked when there would be a ribbon cutting ceremony, Ambassador Johnson said: "At some point we're going to do it, but there's no urgency to that".
"We spend a lot of time talking about the world's problems", he said.
London Mayor Sadiq Khan also said he "would not be welcome" in the city, suggesting there could be widespread protests.
"We treasure this relationship, and I treasure Boris's relationship with me personally and the work that we do together on these many issues".