The new offences will be: Knowingly providing false or misleading information in allowance claims, failing to comply with the rules on registration of interests and breaching the rules which ban paid advocacy - MPs being paid to raise interest groups' issues in the Commons.
Making a false claim would be punishable by up to 12 months in jail or an unlimited fine. The other two would be punishable by fines of up to £5,000.
The Bill would also put the MPs' code of conduct on a statutory footing for the first time and make MPs declare all outside earnings.
Ms Harman told MPs the issue of hiding chunks of published expenses claims would be "looked at again".
Many details were blacked out when full claims for the past four years were published last week - prompting criticism from Freedom of Information campaigners.
Alan Duncan described it as an "unmitigated PR disaster" which "looked like censorship on a massive scale even if it wasn't".
He said it was important to respond to public anger but also to have "a Parliament that works and does not become so brow beaten and rules driven that it loses all the confidence and freedom it needs to do its job properly".
He also questioned whether a move to give more details of MPs' outside earnings was "workable".
Labour MP David Winnick said the blacking out of details, including addresses, delivery people and security claims, had been "very embarrassing".
Ms Harman said the issue would be re-examined, according to the advice of the information commissioner, for the 2008/09 claims.
She said the "fullest possible information compliant with the Freedom of Information Act" should be put in the public domain - but said there were some exceptions, such as telephone and credit card numbers.
Without publishing addresses it was not possible to see if MPs "flipped" second homes - a key part of the Daily Telegraph's revelations from its leaked version of the data.
But it is unlikely MPs' addresses will be included in the next set of publications.
MP Nick Harvey, a member of the supervisory House of Commons Commission, told the BBC the law had been changed to withhold addresses last year and there was no indication from government that that would change.
Earlier, David Heath for the Liberal Democrats also criticised the blacking out of details from claims saying "acres of black space" was a "redaction to the absurd".
He said the plans had his party's "general support" but was concerned about the details of registering non-Parliamentary income.
Gordon Brown told BBC Radio 4's World at One the measures were "the biggest you have ever seen in Parliament" and could be approved by August.
It is a radical change and I think it will fit in with the current public mood
Sir Alistair Graham Former standards chairman
"We are determined to do everything in our power to clean this up and I am not going to rest until we have got this legislation through," he said.
All parties broadly back plans to transfer control of MPs' expenses from the much-criticised Commons Fees Office to the new IPSA.
The plans would allow the IPSA to approve or reject claims and to fine or even expel MPs guilty of serious wrongdoing.
Revelations by the Daily Telegraph about expenses claims made between 2004 and 2008 led to the resignation of a host of MPs and a police investigation into a number of individuals.
The independent Committee on Standards in Public Life is expected to propose permanent changes to the system of MPs' pay and allowances in the autumn.
The committee's former chairman, Sir Alistair Graham, welcomed the government's "pretty draconian package".
He told the BBC: "It is a radical change and I think it will fit in with the current public mood."
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