In Government since March, Lord Harrington stressed his resignation is no reflection on either Liz Truss or Rishi Sunak and he instead feels it is the right time to leave a “temporary” position.
Lord Harrington, who thanked Mr Johnson for appointing him, said his decision means the next prime minister can “save” on a ministerial position.
The former Conservative MP said he had spoken to both Number 10 and Tory leadership candidates Ms Truss and Mr Sunak before publicly confirming his decision.
“We now have the permanent machinery in place to deal with human misery coming to this country from wherever they come from without the need for a designated minister to be in charge,” he said.
“With monthly reports for the Secretary of State, whoever he or she may be, there should now be a permanent delivery mechanism.
“Never again will we have people arrive in the UK and go into hotels without a plan, as happened with those coming from Afghanistan and from the Ukraine, where at the beginning we didn’t have any idea where to put them all.”
He said he will take on a voluntary role helping refugees after he leaves Government.
More than 100,000 Ukrainians have arrived in the UK since the invasion began.
“I was brought in to do a very specific job, which was to set up working machinery across Government to deal with helping the Ukrainians in need,” Lord Harrington said.
“I believe we now have a process and procedure in place that means there won’t necessarily be the need for a minister like myself.
“I’m not walking out on the role or either candidate and will continue to support where helpful. But what I was specifically asked to do is essentially complete so it seems right that I make clear to both leadership contenders that they may be able to save on a ministerial post when they take over.
“Or at least have the option to shape the role for the next phase with a new person now I’ve established a successful system.”
Mr Johnson tweeted on Sunday evening: “Thank you, Richard, for your work on welcoming 120,000 Ukrainian refugees to the UK, following Putin’s war in Ukraine.
“Your wealth of experience in government will be sorely missed.”
Other Tory MPs paid tribute to Lord Harrington on Sunday, with former health secretary Sajid Javid tweeting: “I saw first hand how you did a brilliant job, helping thousands of desperate people. Thank you.”
And Caroline Nokes tweeted: “You worked so hard on this Richard and I have always welcomed your willingness to engage and discuss all the challenges for refugees.”
Mr Johnson, who leaves office on Tuesday, has been praised by Ukrainian President Volodymyr Zelensky.
He wrote a piece in the Mail On Sunday and also spoke to the Sunday Times about his gratitude to the outgoing Prime Minister.
He also expressed his hope for future “close relations” with Mr Johnson’s successor, who could be in line for an invitation to visit Kyiv.
Mr Zelenksy told the Sunday Times that it would be a “priority” to extend an invitation to the next prime minister, while he also admitted he was “concerned” when he heard Mr Johnson was resigning.
“When we learned that there would be a change of government, all of us were concerned,” he told the paper.
“Johnson was supporting us and a lot depends on the leader. The leader is the one who communicates. The leader is the one that mediates between a country and its people, and a leader has an impact upon society.”
Speaking about Mr Johnson’s successor, he said of that relationship: “I can only pray that it will be at the same level as I had with Prime Minister Johnson.”