When it was introduced, it had been expected to apply eventually to peers as well as MPs.
However, as it returned to the Commons on Tuesday, the Lords had been assured "categorically" it would not apply to them.
Two of three proposed offences - relating to breaking rules on the registration of interests or the ban on paid advocacy - had been dropped.
A bid to scrap Parliamentary privilege, which protects MPs from court action over what they say in the Commons, was also rejected after a surprise Commons defeat for the government.
During Tuesday's debate on amendments made in the House of Lords, which were approved without a vote, Mr Straw faced accusations the bill had been rushed through for "public relations purposes" and had since been "emasculated".
He said "improvements" had been made to the bill and the three main parties had backed plans for a new Parliamentary Standards Authority but they had to work "from a standing start".
He said it had been a difficult process and he would have been condemned if he had resisted making changes.
"I think we have achieved a very much better measure as a result," he said.
In a later interview, Mr Straw said the new authorisation process for expenses, combined with full transparency in the disclosure of claims, would mean past abuses could not happen again.
"Thankfully the expenses scandals will be a thing of the past," he told the BBC.
Jack Straw responds to criticism of the legislation
He said any MP "daft" enough to fiddle their expenses in future would be severely punished.
Shadow Commons leader Alan Duncan said the bill had arrived "in a state of some confusion" and he was pleased "significant" concessions had been made.
"It was obvious that ministers had been required at very short notice to create a bill that had to match the prime minister's press release, initially at least, no matter what the consequences."
The Lib Dems said the government's enthusiasm for a radical reform of the political system in the wake of the expenses scandal had been short-lived.
Nick Clegg said it "beggared belief" that MPs should be about to go on an 82-day recess with so little achieved in the way of real reforms to the House of Lords, party funding and new mechanisms for sacking dishonest MPs.
He had called for Parliament to continue sitting until such measures had been introduced.
"It[the bill] is a fig leaf to cover up the government's abject failure to deliver on the promises it made to the British people at the height of the expenses scandal that it would sort out British politics for good," he said.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.