It is shown through three different approaches that, contrary to a longstanding conviction older than 160 years, the orbit of Mercury behaves as required by Newton's equations with a very high precision if one correctly analyzes the situation in the framework of the two-body problem without neglecting the mass of Mercury. General relativity remains more precise than Newtonian physics, but the results in this paper show that Newtonian framework is more powerful than researchers and astronomers were thinking till now, at least for the case of Mercury. The Newtonian formula of theadvance of planets' perihelion breaks down for the other planets. The predicted Newtonian result is indeed too strong for Venus and Earth. Therefore, it is also shown that corrections due to gravitational and rotational time dilation, in an intermediate framework which analyzes gravity between Newton and Einstein, solve the problem. By adding such corrections, a result consistent with the one of general relativity is indeed obtained. Thus, the most important results of this paper are two: i) It is not correct that Newtonian theory cannot predict the anomalous rate of precession of the perihelion of planets' orbit. The real problem is instead that a pure Newtonian prediction is too strong. ii) Perihelion's precession can be achieved with the same precision of general relativity by extending Newtonian gravity through the inclusion of gravitational and rotational time dilation effects. This second result is in agreement with a couple of recent and interesting papers of Hansen, Hartong and Obers. Differently from such papers, in the present work the importance of rotational time dilation is also highlighted.