Robertson said he was "disappointed" that the information could be accessed in this way.
National leader Simon Bridges said Robertson's statement had clearly implied National had hacked Treasury, or had received hacked information.
In a letter, dated May 31, Bennett said the saga raised "serious questions" about the politicisation of the public services – implying that the Government used Treasury to smear the National Party.
Robertson has rejected this accusation.
A spokesman for the State Services Commission (SSC) confirmed on Friday that an inquiry would look into Treasury and issues around its website security.
SSC boss Peter Hughes is considering the allegations made by Bridges around whether Makhlouf smeared National, or misled the Minister of Finance.
At this stage, however, the SSC spokesman said that was not part of the official inquiry.
But Bennett said that needed to be part of the official investigation.
She wants Robertson's role in the saga to be looked into, as well as a complete review of all communications between the Finance Minister's office and the Prime Minister's office under the "no surprises" approach.
What information Treasury and the Finance Minister had at their disposal before they issued those statements also needs to be investigated, Bennett said.
"The actions of the Secretary to the Treasury and the Minister of Finance in misleading the public in this way have had a detrimental effect in maintaining public confidence in a neutral public service that works without political influences," Bennett said in the letter.
"Without a full investigation that covers the issues I raised, it risks derailing the years of progress New Zealand has made as a world-leader in Open and Transparency."
In response, Robertson said: "The SSC is looking into the Treasury's response so the Government needs to wait to see what the result of that inquiry is, but I am hoping there are clear recommendations on how to avoid this kind of situation happening again."