It will continue until "the first bomb drops", he told CNN.
Sanctions and diplomacy, he said, had brought unprecedented international unity against North Korea's nuclear weapons programme.
Last month, Mr Trump told Mr Tillerson not to waste time seeking talks with Kim Jong-un.
In Sunday's interview, Mr Tillerson again refused to comment on whether he had referred to Mr Trump as a moron.
"I'm not going to deal with that petty stuff," he replied, saying he would not dignify the question with an answer.
In recent months, North Korea has defied international opinion by conducting its sixth nuclear test and launching two missiles over Japan.
Analysts say the secretive communist state it is clearly set on developing a nuclear-capable missile, able to threaten the continental US, despite UN sanctions.
Lines of communication
At the end of last month, Mr Tillerson disclosed that the US was in "direct contact" with the North and looking at the possibility of talks.
After months of heated rhetoric, it came as a surprise to some that the two countries had lines of communication.
However, the next day Mr Trump tweeted Mr Tillerson to say: "Save your energy Rex, we'll do what has to be done!"
Mr Tillerson has not denied calling Mr Trump a moron after a July meeting at the Pentagon.
The president responded by challenging the secretary of state to an IQ test but a spokeswoman said later it had been a joke.
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