Anning said Hanson had told the One Nation senators not to do so.
Anning's defection means One Nation has lost three senators in just 16 months.
The One Nation leader then sought to speak to Senator Anning but could only contact his brother, Harry.
Senator Anning raised eyebrows when he was escorted into the chamber on Monday morning by Liberal Democrat David Leyonhjelm and Australian Conservative Cory Bernardi as two of his colleagues Brian Burston and Peter Georgiou sat watching.
Senator Hanson admitted it was an unwelcome distraction that put them behind schedule, but vowed to keep moving.
She said Senator Anning had walked out of a meeting with her this morning while discussing his future, claiming he said, "I know where this is going".
There have been reports of strife between Senator Anning and One Nation and accusations the party asked him to stand aside to return Malcolm Roberts to the Senate.
The move comes after weeks of behind-the-scenes friction between Senator Hanson - a strong backer of Mr Roberts - and members of Mr Anning's incoming staff.
In a statement, Senator Hanson said she asked Senators Burston and Georgiou to mediate with Mr Anning to find some common ground, but they were informed Mr Anning would stand as an Independent only minutes before he was sworn into the Senate. But then the High Court intervened.
Senator Hanson said they would travel to the north Queensland town of Bowen today.
"Senator Leyonhjelm did so with pleasure at the invitation of Senator Anning", said the statement.
He receives 19 first preference votes, but filled the position left vacant by former Senator Malcolm Roberts's being found to be ineligible by the High Court as part of the citizenship scandal.
He had expected to be accompanied by One Nation senators into the Senate chamber for his swearing in, as is usual practice, but that changed at the last moment, he said.
Crossbench Senator David Leyonhjelm Parliament House in Canberra.
The decision made an immediate impact on the Queensland state election campaign, with Premier Annastacia Palaszczuk saying it showed the "chaos" wreaked by One Nation-coalition governments.
Andrew Bartlett and Jordan Steele-John were sworn in as new Greens senators for Queensland and Western Australia respectively.
Today she's writing a new chapter with the defection the party's newest senator, Fraser Anning, the Queenslander meant to replace Malcolm Roberts, who was ousted by the High Court for being a dual national, a problem that had its own freakish twists and turns.
The Anning about-turn is a remarkable move given it is hugely unlikely he would have reached the Senate without the Hanson name on the ballot paper at the double dissolution election past year.
"From that time on, the formalities of completing the process of renunciation were out of their hands and subject to the vagaries of the administrative processes in the United Kingdom", he said.