In this work the influence of the chiral anomaly effect on the evolution of magnetohydrodynamic turbulence was studied. We argue that before the electroweak symmetry breaking and for temperatures high enough such that the electron mass can be ignored, the description of a charged plasma in general needs to take into account the interplay between turbulence and the anomaly effects. It was demonstrated that this generalization can have important consequences on the evolution of turbulence, leading to the creation of maximally-helical fields from initially nonhelical ones. Therefore, chiral effects can strongly support turbulent inverse cascade, and lead to a slower decrease of the magnetic field with time, and also to a faster growth of the correlation length, when compared to the evolution predicted by the standard magnetohydrodynamics description. Using the weak anomaly approximation, and treating the anomaly contributions to magnetic energy and helicity as a small perturbation, we derive the specific solutions for the inverse cascade regime that demonstrate how chiral effects support the inverse cascade.