"We condemn North Korea in the strongest words possible for a reckless act which was a clear violation of United Nations Security Council resolutions," May told a press briefing during an official visit to Japan.
"In response to this illegal action, Prime Minister (Shinzo) Abe and I had agreed to work together and with others in the international community to strengthen pressure against North Korea including by increasing the pace of sanctions."
Britain wants new United Nations sanctions against North Korea that would target guest workers sent mostly to Russia and China, and whose wages are a source of revenue for Pyongyang.
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Last week, Japan expanded its own sanctions against the North, after a similar US move.
May pointed to the key role China plays as the North's chief ally.
"Now we need to ensure it's not just words of condemnation, but that action is taken, and China does have a particular position in this," May said.
"They have leverage on North Korea, and I believe we should be encouraging China to exercise that leverage."
Earlier Thursday, Beijing slammed a report that suggested the US, Japan and Britain were planning fresh punitive measures against the North, saying calls for sanctions were "destructive" and that those measures alone "cannot fundamentally resolve the issue".
May's comments come after she attended Japan's top security meeting Thursday, reportedly only the second foreign leader to attend a meeting of the National Security Council after Australia's then-prime minister Tony Abbott in 2014.
The Council, which was created at the end of 2013, consists of Prime Minister Shinzo Abe and key ministers.
'Like-minded partner'
Before the meeting, May described Japan as "the UK's closest security partner in Asia" and a "like-minded partner".
"We must work together to enhance our collective response to the threats to the international order and global peace and security," she said.
"That must include confronting the threat that North Korea poses and ensuring that this regime in North Korea stops its aggressive acts."
The UN Security Council on Tuesday unanimously condemned North Korea over the launch of the missile, which flew over Japanese territory before crashing into the Pacific, but it stopped short of issuing fresh punitive measures against the regime.
Earlier Thursday, May visited a naval base on the outskirts of Tokyo with Defence Minister Itsunori Onodera before attending a business forum and the security meeting.
My visit "gives a sign of the growing cooperation and partnership that we have on defence", May said of the naval base tour.
Japan's defence ministry on Thursday said it would request its largest-ever annual budget to beef up its missile defence systems in the wake of the North Korean launch.
May arrived in Japan Wednesday with an eye to soothing Brexit fears and pushing ahead on early free-trade talks with the world's number three economy.
She is to meet Emperor Akihito on Friday before leaving.
Earlier this year, Britain and Japan signed a defence logistics treaty that allows both country's forces to share equipment, facilities and services.
