There are several Russian-language sources discussing crime (and punishment) in Odessa in the aftermath of the Russian Civil War (early 1920s):
The Wikipedia article Odessa CheKa, Одесская ЧК, part "Ликвидация бандитизма".
The story of "Murka", История «Мурки».
V.V. Shulgin, "The year 1920. Essays." В.В.Шульгин "1920 год. Очерки."
If one trusts these sources, the regular Odessa criminal police as well as CheKa were too corrupt to handle the situation. As the result, special groups from Moscow CheKa were sent to Odessa to arrest (and kill) the most notorious gangsters. According to Shulgin who lived in Odessa at the time:
"... The Bolsheviks managed very quickly. And we must do them justice, as far as regular criminals are concerned, Odessa soon became quite a safe city..."
Shulgin might be a trustworthy source here since he was a committed anticommunist when he wrote his essays.
The prototype of "Murka" song (mentioned in one of your comments) was Maria Klimova, one of the undercover CheKa agents sent from Moscow to Odessa. Unlike the song's main character, she was not killed by gangsters in 1920, but retired (happily or not, hard to tell) from her job in 1952.