Speaking at a meeting of the U.N. Security Council in New York, Tillerson rebuked the North Korean representative's assertion that the United States is to blame for tensions between the two countries.
Difficulties in obtaining supplies, including medical equipment, and in transferring money to fund aid programmes could have a direct impact on health and nutrition levels throughout North Korea, aid groups say.
Tillerson rejected the argument that the U.S. bore any blame and made clear that there would be no let-up in his campaign to squeeze North Korea and steadily choke off its diplomatic channels and means of financial support.
The US is even more determined to continue its peaceful pressure campaign, Tillerson said, with the goal of "setting the conditions for North Korea to engage in serious negotiations toward the complete, verifiable, and irreversible abandonment of its nuclear weapons programs".
The sanctions are estimated to decrease North Korea's export revenue by $2.3 billion.
US President Donald Trump vowed "it won't happen" after North Korean said in January is was in the final stages of developing a nuclear weapon capable of reaching the United States.
Mr Tillerson's latest message also appeared to place the USA and North Korea at a risky stand-off once more, with Washington insisting on a halt to the North's nuclear weapons and ballistic missile programmes and North Korea advancing them. On Wednesday, a National Security Council spokesman said North Korea must not only first refrain from provocations but also take "sincere and meaningful actions toward denuclearization" for talks to happen.
In October the American president dismissed his Secretary of State's push for talks with the North.
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"China is helping. Russian Federation is not helping". Tillerson also said about full military readiness of the USA due to persistent provocations from Pyongyang, but affirmed that the diplomatic efforts will continue "until the first bomb drops".
"The development and the advancement of those strategic weapons of the DPRK are entirely to defend the sovereignty and territorial integrity of the country from the USA nuclear blackmail policy and the nuclear threat and to ensure the peaceful life of the people", said North Korea's UN Ambassador Ja Song Nam.
He said the US does not want war but "will use all necessary measures to defend itself against North Korean aggression".
Since the spring, the US has run what it calls a "peaceful pressure campaign" against North Korea. "We do not regard this claim as an empty threat".
Singling out China, Pyongyang's sole ally, and Russia, Tillerson urged Beijing and Moscow to go beyond the current tough United Nations sanctions resolutions and impose unilateral measures.
He said "our hope remains that diplomacy will produce a resolution".
But the waves of multilateral and direct U.S. sanctions that have been imposed on Kim Jong Un's regime following its missile launches and nuclear tests have now made operations so hard that some agencies are pulling out.
In his briefing to Friday's meeting, U.N. Secretary-General Antonio Guterres offered his "good offices" to avoid the dispute over the "alarming and accelerated pace" of North Korea's nuclear and missile program accidentally escalation into conflict. Guterres said he was concerned about "unintended escalation or miscalculation", which he said was being increased by "misplaced confidence, risky narratives and the lack of communications channels".