The issue surrounds midfielder Pedro Chirivella, who came off the bench in the 63rd minute of the 2-0 victory, and the absence of an international transfer certificate – required after the 22-year-old returned from a loan spell with Spanish club Extremadura last season.
The English Football League said in a statement: "Following a comprehensive review of all the evidence, the board determined that it wasn't appropriate to expel the club from the competition because of a number of mitigating factors."
Liverpool requested the paperwork in July but the process can only be completed with the Football Association's intervention and that, the PA news agency understands, is where the club believe the error has been made.
EFL rules give it "full powers" to investigate – which could have meant anything from a reprimand to expulsion from the competition.
A statement from the EFL continued: "In particular, 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, 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.
"As a result, the club has also been found guilty of misconduct in relation to the same breach for having fielded the player in the defeat against Oldham in the Leasing.com Trophy on August 7, with the sanction incorporated into the financial penalty.
"The club will be liable for the suspended sum of £100,000, plus an additional sanction, if they were to again field an ineligible player in the Carabao Cup between now and the end of next season."
An EFL spokesman said: "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.
"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 team-sheets despite the lack of clearance. As a result the board concluded the most appropriate sanction was a financial penalty."
Following the EFL ruling, Liverpool issued a statement apologising for the "technical indiscretion".
A Liverpool club spokesperson said: "The club accepts the judgement, outcome and punishment imposed by the EFL.
"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."
When asked about the issue after it came to light, Liverpool manager Jurgen Klopp maintained there would have been nothing untoward behind any administrative error.
"I cannot say too much about it. Other people are working on it. What I can say is if it was our fault alone then probably we need to get punished," the German said.
"It didn't happen on purpose, but my only real concern is the player. He had already a problem, Pedro, that he could not play half a year in Spain and if he were not allowed to play now until it is sorted that would be the biggest problem.
"How it is in life, we make a mistake – I am not sure we really made a mistake – but then it is like this but the player should not be punished for that."