A fight broke out around the hour mark during the match on Tuesday, with Benzarti and Esperance physio Yassine Ben Ahmed the main protagonists.
Police had to intervene to break up the touchline brawl.
Ben Ahmed has also been banned from football for two years, with both men receiving 20,000-dinar fines ($9,155).
Esperance won the Tunisian Cup quarter-final tie 1-0.
Violence is commonplace in football in the north African country, and in May four Stade Tunisien players were handed six-month jail terms for attacking an opponent during a top-flight match.