Mr Sargeant is believed to have taken his own life after he was sacked as Cabinet Secretary for Communities and Children and suspended from the Welsh Labour Party following allegations of inappropriate behaviour towards women.
Family members and friends of Mr Sargeant, who was Assembly Member for Alyn and Deeside, raised concerns that he was not given the opportunity to defend himself and was not made fully aware of the allegations against him before he was suspended by the party.
A statement from Mr Jones, shared from his Twitter account, said that work to establish an inquiry, likely to take place after an inquest, was already underway.
“Further to the First Minister’s comments yesterday about the need for independent scrutiny of his actions and decisions in relation to Carl Sargeant, he agrees that there should be an independent inquiry and it would be proper to ask a senior QC to lead that work.
“To ensure this happens separately from his office, the First Minister has asked the Permanent Secretary to begin preparatory work for this inquiry, and to make contact with the family to discuss the terms and reference of the identity of the QC.
“It is our understanding that such an inquiry should not take place before the outcome of a Coroner’s Inquest – but we will take further advice on this matter.”
Pressure has been mounting on the First Minister to announce such an inquiry since the sudden death of Mr Sargeant last week.
Mr Jones spoke publicly on the matter for the first time on Thursday to say that he had acted "by the book" in the way he had dealt with Mr Sergeant, who he described as a "true force of nature".
After meeting with Welsh Labour Assembly Members, Mr Jones called the situation "the darkest days" for the Assembly - and the "darkest of all" for the Sargeant family.
There had been speculation Mr Jones could resign, but Thursday's speech made no mention of his own future. He said: "There are a lot of inaccuracies in the press and many of you have questions to ask about what happened last week."
Mr Jones precise details "will need to be properly disclosed" at the forthcoming inquest.
"I and my team will of course be cooperating fully with any questions that are raised there," he said. "The family deserve to have their questions answered and if that isn't possible through the inquest then I will endeavour to make that happen through other means.
"I welcome any scrutiny of my actions in the future and it is appropriate for that to be done independently."
Opponents of Mr Jones have rounded on him to criticise his handling of the situation, which emerged in the wake of the Westminster "sleaze dossier". Mr Attridge also alleged that Mr Sargeant had been the target of bullying in the Welsh government.
Opponents of Mr Jones have rounded on him to criticise his handling of the situation, which emerged in the wake of the Westminster "sleaze dossier".Mr Sargeant's lifelong friend and Flintshire council's deputy leader Bernie Attridge, called for Mr Jones to resign saying he "had not done the decent thing".
Mr Attridge also alleged that Mr Sargeant had been the target of bullying in the Welsh government.