Instead the issue will be discussed on 24 November in Parliament’s second chamber, Westminster Hall, where there are no votes.
Conservative Dominic Raab, who led the call, said the result was “a glass-half-full situation”.
Critics of the current extradition arrangements say they are biased against British citizens.
They argue it is unfair for the United States to require “sufficient evidence to establish probable cause” before agreeing to extradite anyone to Britain, while Britons are denied the same protection.
They also disapprove of the European Arrest Warrant system (EAW), which allows fast-track extraditions on the assumption that standards of justice are adequate across Europe.
But earlier this month an independent review, led by former Court of Appeal judge Sir Scott Baker, concluded that the arrangements with the US were not unbalanced.
He also found that the EAW did not operate “unfairly or oppressively”.
Mr Raab applied to the Commons backbench business committee on Tuesday for a full debate on the subject.
He said it was “high time we stood up to ensure proper safeguards for our citizens”.
But his motion was rejected in favour of a debate on cutting fuel prices, sought by Conservative Robert Halfon and backed by a online petition with more than 100,000 signatures.
Speaking after the decision, the Esher and Walton MP said: “Parliament will have an opportunity to voice its concerns, while the government is considering the Baker review.
“Depending on how long those deliberations last, there is a reasonable chance we may secure a slot in the chamber on a voteable motion early in the New Year.
“They key thing is to make sure we don’t lose this opportunity to press for proper reform – under both the UK-US treaty and the European Arrest Warrant. “
Mr Raab had the backing of former Liberal Democrat leader Sir Menzies Campbell and the Labour chairman of the Home Affairs Committee, Keith Vaz, for his extradition motion.
Mr Halfon, whose fuel price debate will take place on 15 November, said: “This is the most pressing issue for the country – it’s causing misery for families, misery for businesses and it’s creating a poverty trap.
“It’s also the biggest break on growth in the economy because businesses are going bust because they can’t afford the price of petrol.”
Home Secretary Theresa May is currently considering the findings of the Baker report on extradition.
Mr Raab and his supporters want her to implement a series of reforms based on recommendations made by Parliament’s Joint Committee on Human Rights earlier this year.
They include increasing the proof required to extradite a British citizen to the US and seeking changes to European legislation to fix a number of problems with the EAW.
The Baker review followed a number of high-profile cases – including that of Gary McKinnon, the alleged hacker who has been fighting extradition to the US for years.
Janis Sharp, the mother of Mr McKinnon – who has Asperger’s syndrome – said she was “shocked” by its findings.
The report prompted criticism from a number of other sources too, including Mr Vaz, who said he was “concerned” that it had “not shed any light on the ease of extradition”.
Human rights organisation Liberty said it did not just disagree with the review, but was “completely baffled” by it.
But Sir Scott said much of the criticism aimed at the Extradition Act was based on a “misunderstanding” of how it worked in practice.