We report on experimental observations of self-sustaining convective patterns in a dusty plasma system. The experiments are carried out in a capacitively coupled radio frequency (RF) argon plasma discharge in which a dusty plasma is produced by introducing micrometer-sized melamine formaldehyde particles. A self-sustained pair of convective cells with opposite vorticities is seen to appear in a lateral view of the system, beyond some threshold values of the background gas pressure and RF power. Particle tracking velocimetry measurements reveal the existence of a dust temperature gradient that is responsible for the convective counter-rotating patterns. The origin of the kinetic temperature gradient in the absence of any external heat source is attributed to the larger ion flux induced heating of the bottom layer of the dust cloud. The velocity and vorticity of the particles in the convective cell increase with an increase in the vertical kinetic temperature gradient. The experimental results compare reasonably well with the findings of molecular dynamics simulations of the laboratory system.