"Yesterday, as a result of an administrative process failure, the government senators in the chamber ended up, on advice, voting in support of the motion", he said.
Senator Cormann, who is leader of Government business in the Senate, did not give details on the nature of the error, but said he was embarrassed and regretful about the error and took full responsibility.
The United Australia Party recruit's motion referring Senator Hanson to an inquiry will investigate whether there was improper interference with the performance of his duties as a senator or if he was penalised for his conduct as a senator.
In response to questions on Tuesday morning, Cormann repeatedly said it was an "error" and that the government had meant to vote against Hanson's motion.
It emerged on Tuesday that Attorney General Christian Porter had issued instructions to senators from the governing Liberal Party to back Hanson's motion.
The vote was surprisingly tight, with Australia's first parliamentarian of black African descent, Lucy Gichuhi, supporting Ms Hanson's motion, but it was ultimately defeated by 31 votes to 28 - rather implying that it is perhaps officially not okay to be white in the former British colony.
"It appears that, of the very large number of motions on which my office's views are routinely sought, this one was not escalated to me because it was interpreted in my office as a motion opposing racism".
Following the vote, Porter tweeted: "The Government Senators' actions in the Senate this afternoon confirm that the Government deplores racism of any kind". Had it been raised directly with me those issues would have been identified'.
He repeated Cormann's statement that it was an "administrative error", and he is reviewing the processes in his office.
"This is a phrase used by white supremacists", Labor senator Penny Wong told reporters.
Being asked to vote on 50 to 60 motions a week - when it's your job that you're paid well to do - shouldn't be that hard.
Senator Derryn Hinch, who voted against the motion and said it "could have been written on a piece of toilet paper", wasn't having it.
A closer examination of the audio captured in the Senate chamber during the vote has picked up a government senator seemingly bragging that he didn't know what he was voting for.