Their main criticism concerns an incident in the second half of the quarter-final first leg in Paris in which Javier Mascherano and Jordi Alba collided and were both lying prone on the ground in the penalty area.
"The club has written to UEFA due to a series of incidents on the pitch in the understanding that objectively speaking it was evident that the referee did not apply the rules of the game," Barca spokesman Toni Freixa told a news conference after a meeting of the club's board on Thursday.
"When two players from the same team are on the floor the referee should stop the match, which did not happen.
"What's more he then obliged them to leave the pitch, in contravention of the rules.
"An incident that is sufficiently serious in a competition like the Champions League to bring to UEFA's attention so that it does not happen again."
Barca were also incensed that Zlatan Ibrahimovic's goal to make the score 1-1 in the 79th minute was allowed to stand when the Swede was clearly in an offside position.
Ibrahimovic, who spent a season at Barca in 2009/10, was playing only because UEFA halved on appeal the two-match ban he received after his sending-off at Valencia in the last 16 round.
"We are fully conscious of the possibility of errors in refereeing and we do not consider ourselves badly treated," Freixa said.
"But it surprises us that in a high-level competition a referee does not know the rules.
"We are not judging the work of referees but the basic principle that a judge should know the law. We hope it does not happen again."
Barca host PSG in the return game at the Nou Camp on Wednesday.
