On Saturday, ESPN's Field Yates reported that Branch will be able to remain on the field after winning his appeal.
Branch's agent later confirmed the news on Twitter. The 31-year-old, who is in the final season of his contract, has one sack, one forced fumble, one blocked field goal, six quarterback pressures, eight run stuffs and a drawn holding penalty.
Nick Shook of NFL.com noted Branch was notified of the possible ban November 21 for a violation of the league's policy on substances of abuse.
Branch had been notified in late November that he was facing a four-game ban for violation of the league's policy on substances of abuse.