Pedro Chirivella played in that match, a 2-0 Liverpool win, despite having not received international clearance after the expiry of his loan last season with Spanish club Extremadura Union Deportiva.
There were fears that the Reds would be expelled from the competition after the EFL opened an investigation into the matter last week.
However, the EFL has ruled that Liverpool would only be fined, not expelled, "because of a number of mitigating factors."
"The board noted that the club had sought the assistance of the Football Association in securing the return of the international clearance prior to the start of this season," an EFL statement read.
"And the club had been able to include the player on team sheets for Premier League Two matches (under Premier League Rules) and one Leasing.com Trophy match, which resulted in the breach not being identified until the club reported the issue to the EFL and Premier League following the MK Dons match."
Liverpool may not even have to pay the full £200,000 fine, as half of that amount has been suspended and will not be enforced unless the Reds field an ineligible player in the Carabao Cup between now and the end of next season.
The Reds indicated in a statement that they were comfortable with the punishment handed out by the EFL.
"The club accepts the judgement, outcome and punishment imposed by the EFL," the statement read.
"We believe it is proportionate with the technical indiscretion committed and will be making no further representations.
"Even though there were mitigating factors, which were beyond our control or jurisdiction, we believe it appropriate we apologise to the competition's governing body and also to Milton Keynes Dons."
With Liverpool now eligible to continue in the competition, they will take on Arsenal in the fourth round, which will take place in the week commencing October 28.
“The rules of the Carabao Cup require clubs to ensure all players are registered and eligible in line with their respective League’s rules before they play," an EFL spokesman said.
"The player was always contracted to Liverpool FC during this period, and the reason he was not eligible was because the club did not have the relevant international clearance following expiry of an earlier loan agreement.
"The club’s breach was in part due to the challenges it encountered with securing the correct international clearance, and its subsequent ability to include the player on teamsheets despite the lack of clearance.
“As a result the board concluded the most appropriate sanction was a financial penalty.”