Barcola completed his move to Anfield late in the transfer window, signing a contract running until the summer of 2031 in a deal reported to be worth between £123m and £145m depending on how add-ons are calculated.
Speaking about the decision to let the France international leave, PSG head coach Luis Enrique said the player's desire for a bigger role simply could not be squared with the realities of squad selection at the Parc des Princes.
"His wish was to play a more important role in the team, something I could not guarantee him," Enrique said.
"In the end, I think everyone benefits from this situation. I accept it and I wish him all the best."
Of the 49 competitive appearances Barcola made for PSG last season, 14 came from the bench, a level of involvement Enrique acknowledged was ultimately not enough for the winger.
"He wanted to leave because he wanted more minutes and a bigger role in the team," the head coach said. "And that is something I cannot guarantee any player, because you have to earn it in training and in matches."
Despite losing a player who scored 13 goals and added seven assists last season, the 56-year-old was adamant PSG would not be worse off.
"In my opinion, we have had an exceptional transfer window," Enrique said. "Without a doubt." The club are also reported to be closing in on a new contract for Ousmane Dembélé, though it remains unconfirmed; the 29-year-old's current deal runs until the summer of 2028.
Barcola's exit brings to an end a spell in Paris in which he established himself as a key attacking option, having joined from Lyon and gone on to win multiple domestic and European honours with PSG, including the Champions League.
He now joins a Liverpool squad already reshaped by a host of attacking arrivals this summer, with the club hoping his pace and directness can add a fresh dimension to their forward line as they look to build momentum after a mixed start to the Premier League season.