"The only message I have, without ... taking the place of the Church, of the judges -- because an investigation is under way -- is that he must take responsibility, speak and act," Valls told the national TV station BFM TV.
"I expect not only words, but acts," the prime minister added.
Barbarin is one of several senior officials being sued for the alleged cover-up.
He is accused of being aware that priest Bernard Preynat had been sexually abusing young boys between 1986 and 1991.
According to a statement from the dioceses of Lyon, the complaint that was lodged in February was made by a highly placed government official, aged 42, who is said to have developed suicidal tendencies as a result of the abuse.
February’s complaint follows an earlier complaint lodged in 2009. No action was taken over the complaint because the events it related to were considered too old.
In an interview with the French daily Le Figaro this morning, the victim, who is simply identified as Pierre, said he was prompted to come forward following allegations made in other cases in the same dioceses.
Earlier this month, a French judge has opened a preliminary investigation into "failure to report a crime" after the alleged victims of a pedophile priest said top Catholic officials covered up for him.
The victims alleged that senior figures in the diocese of Lyon, in eastern France, including Archbishop Philippe Barbarin failed to report the priest, who has been charged with sexually abusing minors between 1986 and 1991, to police.
A formal investigation was opened on January 27 after Bernard Preynat, the priest, admitted he sexually abused young Scouts over 25 years ago.
His lawyer, Federic Doyez, said Preynat told the judge that "the facts had been known by the church authorities since 1991", when he was expelled from the independent Scouts group he had led for nearly 20 years.
In an interview earlier this year, Barbarin, 65, said he had been made aware of the priest's behaviour "around 2007-8" and the Vatican earlier gave the cardinal its backing, saying it had confidence he would deal with the matter "with great responsibility".