The Reds were the dominant side throughout the Friday night match; however, their attacks were met with a solid Saba defense in the first half.
Zlatko Kranjcar's men had to wait until the 80th minute for the breakthrough, which came when a magnificent build-up play led to a Karim Baqeri strike.
It was the first time this season -- after a three-game ban imposed by Iran's Premier League (IPL) Disciplinary Committee -- that Persepolis was allowed to have spectators.
The 1-0 victory was Persepolis' second win in a row followed by two consecutive draws against Mes Kerman (3-3) in the season opener and Esteqlal Ahwaz (1-1).
The Reds have now moved to fourth place joining the title race also contested by Zob Ahan, Steel Azin, Moqavemat and arch-rival Esteqlal.
Zob Ahan currently sits atop Iran's Premier League with 10 points followed by Steel Azin which has 9 points. Moqavemat, Persepolis and Esteqlal all have 8 point with Moqavemat in third place on goal difference.