The Spanish champions went ahead in the first leg clash through a Lionel Messi penalty nine minutes into the second half, but replays suggest Martin Demichelis' challenge on the forward, which saw him get sent off, may have happened just outside the area.
Daniel Alves doubled the advantage in the closing stages at the Etihad Stadium, but Pellegrini took umbrage with the performance of the Swedish official.
"Of course I have a complaint, for the whole match I feel we didn't have a referee who was partial for both teams," he told Sky Sports. "It was not a good idea to put a referee from Sweden in charge of such an important game.
"The referee was three metres from a foul (on Jesus Navas in the build up to the penalty decision), he didn't give it and it was a foul outside the box (for Demichelis' tackle on Messi)."
Eriksson previously endured the wrath of Barca captain Carles Puyol and former boss Pep Guardiola following a Champions League quarter-final clash with Milan in 2012, and Pellegrini believes his appointment for this fixture was an error.
"I think it was a mistake to nominate a referee who had previously had problems with Barcelona in the past," he continued. "We had the same referee and once again he made mistakes."
Meanwhile, City captain Vincent Kompany believes his side can feel hard done by.
"2-0 is very tough on us," he told ITV. "When it was 11 v 11 we were in the game but with a man less and with their individual quality which is so high it was very hard.
"If they play as they can in the second leg it will be very hard but we still have a chance."
