Mr Brafman also said he would investigate what he said had been "perjury" before the grand jury which indicted Mr Weinstein, as well as the conduct of a NY police officer who had worked on the case.
"The integrity of these proceedings has been compromised", he said.
It leaves Weinstein facing allegations largely related to two women, but Brafman argued that Evans' "perjury" also taints the first count against Weinstein, if not the entire case.
"Sexual assault is a serious crime but falsely accusing someone of sexual assault is also a serious crime", Brafman said outside court on Thursday.
The attorney said her client will continue her fight in other venues.
US-based tabloid website TMZ reported last week that a casting director claimed Evans once told her she performed oral sex on Weinstein to gain an acting role.
Weinstein's lawyers and prosecutors had been wrangling over the part of the indictment pertaining to Evans in recent weeks in closed-door meetings and sealed court papers.
During the Thursday hearing, the prosecutors agreed to drop charges relating to an alleged sexual assault.
The development was announced in court with Weinstein looking on.
Weinstein's lawyer, Ben Brafman, has said in court filings that prosecutors withheld evidence that would have made the grand jury think twice about charging him, such as friendly emails one accuser sent after the alleged rape.
The 66-year-old father of five, out on a $1 million bail and forced to wear a Global Positioning System monitor, has long pleaded not guilty to six counts allegedly committed against three women in 2004, 2006 and 2013.
In a shocking disclosure, the pal said that she gave the same account to the detective during the February 2 phone interview while her attorney was present - adding that she had also told him that a fact checker from the New Yorker had called to confirm Evans' assault claims. Weinstein had promised to get her an acting job if she agreed to perform oral sex and she agreed.
She says Weinstein forced her into oral sex in 2004.
The detective has been removed from the case, Brafman said. Rather, "it only speaks volumes about the Manhattan D.A.'s office and its mishandling of my client's case".
Mr Weinstein has been accused of sexual misconduct by more than 70 women. "The evidence shows that the criminal case against him is strong".
Evans' lawyer, Carrie Goldberg, furiously said outside court that her client had been abandoned by Manhattan District Attorney Cyrus R. Vance Jr. for no reason. At the time, Vance cited a lack of supporting evidence, despite the existence of a clandestinely made recording of Weinstein discussing the episode with the woman. Weinstein, 66, has pleaded not guilty and is free on $1 million bail.
New York Police officials poured on the pressure, too, saying publicly they believed they had gathered ample evidence to make an arrest.