Jeremy Quinn admitted: "I don't deny that we've got serious issues that we need to resolve," and also went into greater detail about the reports of noise and vibrations on the vehicles.
With the Ajax project also delayed, the programme's cost was also called into question.
Shadow Defence Secretary John Healey asked Jeremy Quin during House of Commons Urgent Questions on Defence "what progress has been made" on the Ajax programme.
"£3.5bn paid out, four years late and just 14 vehicles delivered, light tanks that can't fire while moving and vehicle crews made so sick that the testing has been paused," he said.
"If this was defence procurement the minister is content is broadly on track, how badly has it got to go wrong before he'll admit the contract is flawed?
"Ministers are failing British Forces and failing British taxpayers," he added.
Mr Healey also said Ajax "requires independent scrutiny", and asked if the National Audit Office (NAO) would be doing "an urgent special audit" and whether the Government would "now halt the plans to cut Army numbers and focus instead on fixing this failing procurement system".
Mr Quin responded by admitting he would "not hide" that the Ministry of Defence (MOD) has "two primary concerns: noise and vibration".
"On noise, we have mitigations currently in place to enable a certain element of training, albeit reduced training," he said.
Watch: One defence expert described the Ajax programme as a 'fiasco' last week.
"We were very concerned by reports of noise issues in the vehicle.
"All personnel who may have been exposed to excessive noise have been tested and training was paused.
"It now continues with mitigations in place as we pursue resolution.
"We're looking at two headsets which hopefully within the next few weeks, they will be approved for use which will further extend what we can do in terms of training.
"But, that does not get us to the root cause and we need to get to the root cause of the noise issues within this vehicle, be that mechanical or, indeed, electronic."
Mr Quin said the MOD was "concerned" about the excessive vibration, despite General Dynamics, the prime contractor on the Ajax programme, not having "had the same experience of vibration".
He added he "absolutely trust[s] the reports that have come to me by our service personnel".
"We have also commissioned independent vibration trials from world class specialists at Millbrook Proving Ground which should conclude next month," he said.
Mr Quin also addressed the four-year delay in delivering the Ajax programme.
"Maingate One approval was granted in March 2010," he said.
"Negotiations with the prime contractor to recast the contract were held between December 2018 and May 2019.
"The forecast initial operating capability, the IOC, was delayed by a year to 30 June 2021, later this month, at 50% confidence – with 90% confidence of September 21.
"Despite the ongoing impact of COVID, we have stuck by this IOC date – though it remains, of course, subject to review.
"By the end of next week, we will have received the requisite number of vehicles to meet IOC, the necessary simulators have been delivered and training courses commenced.
"These delivered vehicles are all at capability drop one standard, designed for the experimentation, training and the familiarisation of crews first in line for the vehicles.
"Capability Drop Three, applying the lessons of the demonstration phase, is designed for operations.
"We remain in the demonstration phase, and as with all such phases, issues have emerged with the vehicle that we need to resolve.
"I can assure the House that we will not accept a vehicle that falls short of our requirements and are working with General Dynamics, the prime contractor, to achieve IOC.
Mr Quin also addressed Mr Healey's question regarding the price of the programme, outlining the agreement with General Dynamics is a "firm price contract".
"So, £5.5bn is the maximum that is payable, including VAT," he said.
"Currently we're at... just under £3.2bn spent."
"There is a heavy incentivisation on our suppliers to ensure they get this over the line."
The defence minister also addressed claims Ajax is unable to fire on the move, or able to travel above 30km per hour.
"I can reassure [Mr Healey] that the vehicle is capable of going well ahead of 30km per hour but, with newly trained crews, a certification has been placed restricting speed," Mr Quin told the Commons.
"I would expect that to be lifted during the course of next month."
He also said there was a restriction in terms "of going up over a reverse step".
"This is a vehicle capable of going over a 75cm object, reversing over it.
"A restriction has been placed, I would expect that to be lifted shortly too.
"This is a vehicle that is capable of firing on the move. It is not something that we have certified it to do as yet."
He added that Ajax was working "through what is a demonstration phase, but we will continue to advance that demonstration phase".
Mr Quin closed his defence of the programme by stating Ajax is a "first-class vehicle".
"It is the first of its kind," he said. "It's got an important job to do.
"We will and we must get this right, and get it delivered," he added.
Cover image: Ajax vehicle pictured during a demonstration at Salisbury Plain Training Area (Picture: MOD).