The Public Accounts Committee said the new Personal Independence Payment scheme had been "rushed" through, with a "shocking" impact on claimants.
"Many" faced six-month delays, with terminally ill people waiting a month on average for the payment, it said.
Ministers defended the system and said the MPs' figures were out of date.
Minister for Disabled People Mike Penning said the new PIP system ensured "support goes to those who need it most".
He insisted the report was "based on old statistics", adding: "I have introduced a faster process for people with terminal illnesses, with clearance times reducing to our target of 10 days.
"And a higher proportion of people are getting the highest rate of financial support for daily living under Personal Independence Payment than DLA."
New claims for the Personal Independence Payment (PIP) - which replaces the Disability Living Allowance (DLA) - began in April 2013. They are worth between Â£21 and Â£134 a week.
Most people applying for PIP have a face-to-face assessment to determine eligibility, which have been carried out by the private contractors Atos Healthcare and Capita Business Services.
In its report, the cross-party committee accused Atos of providing "incorrect and potentially misleading" information about its capabilities when bidding for the contract to conduct assessments for the government.
The company said it completely refuted the claim.
The Department for Work and Pensions began processing new claims for PIP in northern England in 2013, but had only made 360 decisions when the scheme was introduced nationwide in June.
Reassessment of the existing 1.7 million DLA claimants began in October, but was effectively paused after a backlog of some 780,000 claims built up.
Committee chairwoman and Labour MP Margaret Hodge said: "The implementation of the Personal Independence Payment has been nothing short of a fiasco.
"The Department for Work and Pensions has let down some of the most vulnerable people in our society, many of whom have had to wait more than six months for their claims to be decided.
"The department's failure to pilot the scheme meant that the most basic assumptions, such as how long assessments would take and how many would require face-to-face consultations, had not been fully tested and proved to be wrong."
She continued that this had resulted in "significant delays, a backlog of claims and unnecessary distress for claimants who have been unable to access the support they need to live, and in some cases work, independently".
"The personal stories we heard were shocking," Mrs Hodge added.
In one case, a claimant required hospital intervention as a result of the stress caused by the delays.
Elsewhere, claimants had been forced to turn to food banks, loans and charitable donations to support the extra costs of living associated with their disability, the MPs said.
The committee also found the average waiting time for terminally ill people to receive a decision was 28 days - 180% longer than originally expected.
Claimants had also endured an "unacceptable" standard of service, with assessors failing to turn up at centres or cancelling home visits at the last minute.
"We are concerned that Atos appears to have included incorrect and potentially misleading information in its bid for the contract," Mrs Hodge went on.
"Atos stated in its tender document that it had 'contractual agreements' in place with a national network of 56 NHS hospitals, 25 private hospitals and over 650 physiotherapy practices to provide assessments.
"This turned out not to be true."
Kate Green, Labour's shadow disability minister, said: "This scathing criticism of the government's chaotic handling of PIP leaves serious questions about competence at Department for Work and Pensions.
"Thousands of disabled people are waiting months and months for essential help which means taxpayers are facing a huge bill to deal with the enormous backlog of Personal Independence Payment assessments."
An Atos Healthcare spokeswoman said: "We completely refute any allegation of misinformation during the procurement process for Personal Independence Payment.
"The Department [DWP] made clear that they were not misinformed during the tender process, that at the point of 'go live' they knew our capacity, our partners and the number of centres we would be using."
She added Atos had written to the committee to clarify its position and had "invited the National Audit Office in to scrutinise our documentation".
In a separate development, leaked government documents suggested Atos's PIP assessment backlog would not be cleared until March next year.
According to the memos, Capita's backlog should be cleared by the end of November thanks to an additional Â£27m the company has been paid by the government.
Changes to benefits for sick and disabled people have led to a "fiasco", causing unnecessary distress to thousands of people, MPs have warned.