Both of Kweuke’s strikes came in the opening 45 minutes as Karadeniz Atmacasi took a 4-0 lead into the break.
The skipper first headed home in the seventh minute from Suleyman Koc’s corner after losing his marker and powering home from the near post.
He then doubled Rize’s lead after some brilliant work from Dhurgham Ismail, who skipped past a few challenges before crossing for the towering talisman to head home into an empty net from close range.
The strikes were the 29-year-old’s 11th and 12th of the campaign as he helped earn the three point for the relegation-threatened side.
Not content with just scoring goals, Kweuke turned provider when he set up Nigeria's Bright Edomwonyi for his first Super Lig goal for his side.
Aboubakar bagged his ninth of the campaign when he opened the scoring at the stroke of half-time, but an own goal in stoppage time saw Fenerbahce fight their way back to salvage a hard-earned draw.