The blunder sent shockwaves across the country after a member of the public posted a photograph from the event on social media.
Town hall chiefs in Santa Fe decided to mark July 9 by hoisting the national flags of all the countries on the American continent that had existed since the main square was built in 1973 without realising that the Falklands flag was among them.
Interim mayor Juan Manuel Mansilla said: “It was a protocol and ceremonial error. First of all, we want to say sorry.
“It was an error of the Protocol Directorate. It was reported that we hoisted the British flag when it was not like that.”
He continued: “The national flag was hoisted. The Directorate of Protocol requested the flags that existed when this was built in 1973 and did not notice that within the 24 flags, this one was present.
“It was an involuntary mistake, an oversight, which should certainly not happen again.”
Santa Fe officials have also issued a statement apologising to veterans of the 1982 military campaign to take control of the islands the Argentinians know as Las Malvinas.
The statement said: “We apologise to our Malvinas heroes, we recognise them, we love them and respect them.
“We regret this situation, which surely led them to remember those dark years, when the Argentine people did not recognise them as they should have and as we do today.
“Once again, sorry, an error that must not be overlooked and for which the corresponding measures will be taken”.
The Falklands conflict erupted on Friday 2 April 1982 when Argentina invaded and occupied the British territory in the South Atlantic.
Three days later Margaret Thatcher dispatched a naval task force to engage the Argentine Navy and Air Force before making an amphibious assault on the islands.
The conflict lasted 74 days and ended with the Argentine surrender on 14 June 1982, returning the islands to British control.
In total, 649 Argentine military personnel, 255 British military personnel, and three Falkland Islanders died during the hostilities.