Pope Innocent XI elevated him into the cardinalate on 2 September 1686 as the
Cardinal-Priest of Santa Susanna. He was transferred to the see of
Montefiascone and Corneto in 1687 with the title of Archbishop. He established an orphanage in Corneto which was subsequently named "Orfanotrofio Barbarigo" in his honor. From 1692 to 1694, Venerini opened ten schools in Montefiascone and the villages surrounding
Lake Bolsena. The cardinal rented a house for Venerini and her school, and provided the material means; Rosa trained the teachers, and organized the schools. When she had to return to Viterbo to attend to her first school, Venerini entrusted the Montefiascone schools and the teachers to the direction of a young woman,
Lucy Filippini, whom the cardinal had entrusted with the leadership of the project in his diocese. Filippini organized the teachers of that diocese as a separate
religious congregation known as the
Religious Teachers Filippini. He participated in the papal conclave of 1689 which resulted in the election of
Pope Alexander VIII and also partook in the conclave of 1691 that saw the election of
Pope Innocent XII. He later opted to be the
Cardinal-Priest of San Marco in 1697 and partook in the papal conclave of 1700 that saw the election of
Pope Clement XI. Barbarigo died on 26 May 1706 at 10:00am in Montefiascone in the residence of the bishop. He was buried in the tomb where bishops of the see were buried but was later transferred to another church and his heart placed in the Montefiascone
seminary. ==Sainthood==