The current research proposes a novel method for printing continuous carbon fiber composite parts. At first, continuous carbon fiber prepreg filament for Fused Deposition Modeling 3D printing was manufactured, followed by modification of extruder head of 3D printers to print the filament. Thereafter, three-point flexural test and Response Surface Methodology were adopted to study the mechanical properties of the composite parts printed with the filament. After testing, a mathematical model was developed to describe and analyze the relationship between the printing parameters (printing temperature, printing speed, and layer thickness) and the flexural strength of printed composite parts. We discovered that the flexural strength and flexural modulus of printed composites significantly improved with the proposed method with specified printing parameters, and while of all the parameters, the layer thickness had the greatest contribution towards the final flexural strength. The results indicate that the discussed method could be a promising approach to print CCF composites.