The late time crossover from a power-law to an exponential expansion of the Universe evolution is the major problem in today’s physical cosmology. Unless this critical transition problem is solved, it is not possible to reach a holistic theory of cosmology. In this study, we propose a simple model in the FLRW framework, where dark matter and dark energy interact through a potential. We analytically solve this model and obtain scale factor a(t) from the presented model. Mainly, employing numerical solutions we show that the scale parameter has a hybrid form which includes power and exponential terms. The numerical results clearly show that there is a time crossover tc in the scale factor a(t) curve, which indicates the transition from the power-law to the exponential expansion of the Universe. We ﬁt these unscaled curves and obtain that scale factor behaves as a(t) ∝ t2/3 below t ≤ tc, and as a(t) ∝ exp(H0t) with H0 = 0.4 and H0 = 0.3 for the relatively weak and strong interactions above t > tc, respectively. It is the ﬁrst time that we explicitly obtain a hybrid scale factor incorporating the power and exponential terms as a(t) ∝ t2/3eH0t . We conclude that the presented model can solve the late time transition problem of the Universe based on dark matter and dark energy interaction. Additionally, we numerically obtain other kinematic parameters depending upon the scale factor. We discuss the limit behaviors of all relevant cosmological parameters. Our results are completely in good agreement with observational data. Finally, we state that this work makes essential steps towards solving a critical outstanding problem of the cosmology, and has a potential to creates a paradigm for future studies in this ﬁeld.