Striker Zafer Biryol scored twice in the 12th and 25th minutes before Brazilian forward Tita Dos Santos, on loan from Ankaraspor, pulled one back in the 79th minute.
Altan Aksoy sealed the victory for the hosts two minutes from time to boost their chances of staying in the top flight next season.
Egyptian defensive duo Ibrahim Saied and Beshir Al-Tabei were included in the starting line-up and put on good performance.
The result lifted Rizespor to the 15th place in the league table with 35 points while Ankaragucu lie 13th on 36 points.
Elsewhere, Konyaspor were handed a 2-1 away defeat at the hands of Kayserispor.
Dimitar Ivankov grabbed the opener as early as the fifth minute and Gokhan Unal doubled the lead in the 23rd minute while French striker Cedric Sabin notched the consolation for Konyaspor in the last minute.
Abdul-Zaher Al-Saqqa and former Ahli right-back Mohamed Abdullah took part in the game.
Konyaspor are eighth in the table with 39 points as their hopes of securing a UEFA Cup spot almost demolished.