In a move aimed at dampening the impact of “broken windows” policing, the New York City Council on Wednesday passed bills to create a civil process for some of the most common low-level infractions observed by police officers, including public drinking and public urination.

The passage of the bills, known collectively as the Criminal Justice Reform Act and spearheaded by the Council speaker, Melissa Mark-Viverito, was the most significant step yet toward reducing the burden of a two-decade-old policing policy that treats public disorder as harbingers of more dangerous offenses, and has resulted in hundreds of thousands of outstanding criminal court warrants for minor infractions.

“Lives and trajectories will be changed because of what we are set to pass today,” Ms. Mark-Viverito, a Democrat, said shortly before the vote.

The eight bills, introduced in February, passed with strong majorities in the 51-member Council and have the support of the Police Department, which preserves the power to arrest offenders if necessary, and of Mayor Bill de Blasio, who called them “crucial” to the goal of “building a fairer criminal justice system.”