The Education and Skills Select Committee said that both organisations were deficient in the way they ran the flagship individual learning accounts programme, which were shut down last year amid fears of widespread fraud and overspending.
Poor security in the Capita-run computer systems and the lack of accreditation for training providers in the scheme, made it easy for unscrupulous firms to claim subsidies for training they had not given or for students that did not exist, the committee said.
The extent of the fraud is still unclear, but could run into millions. Nine police forces are investigating 80 learning providers and the education department's special investigation unit is holding discussions with a further 16. Ten people have been cautioned and 13 charged.
The MPs criticised the education and skills department, for agreeing a contract with Capita, which did not lay down clear responsibilities for the outsourcer, and left the department with little room to manoeuvre when things went wrong, the MPs said.
"The government was hiring a leading company to run its computer systems. The contract should have been watertight," said committee chairman, Barry Shearman.
Although the government sought advice from KPMG on the design of the scheme, the final specification was deficient and did not hold Capita to account, the MPs said.
Capita should have warned department officials about the security weaknesses of the system, and should have realised that the ILA scheme was a "disaster waiting to happen".
Even when the contract laid down clear goals, such as handling complaints from learning providers and members of the public, Capita did not perform well.
"We were critical of Capita for some of the deficiencies in the quality of advice they were able to give to the department and some clear failings in their contract, and for example, the complaints procedure where Capita should have given a better service," said Shearman.
The failures of the ILA programme will have badly damaged Capita's credibility, said MP Jeff Ennis.