A new laser-control scenario of unidirectional π-electron rotations in a low-symmetry aromatic ring molecule having no degenerate excited states is proposed. This scenario is based on dynamic Stark shifts of two relevant excited states using two linearly polarized stationary lasers. Each laser is set to selectively interact with one of the two electronic states, the lower and higher excited states are shifted up and down with the same rate, respectively, and the two excited states become degenerate at their midpoint. One of the four control parameters of the two lasers, i.e. two frequencies and two intensities, determines the values of all the other parameters. The direction of π-electron rotations, clockwise or counter-clockwise rotation, depends on the sign of the relative phase of the two lasers at the initial time. An analytical expression for the time-dependent expectation value of the rotational angular momentum operator is derived using the rotating wave approximation (RWA). The control scenario depends on the initial condition of the electronic states. The control scenario with the ground state as the initial condition was applied to toluene molecules. The derived time-dependent angular momentum consists of a train of unidirectional angular momentum pulses. The validity of the RWA was checked by numerically solving the time-dependent Schrödinger equation. The simulation results suggest an experimental realization of the induction of unidirectional π-electron rotations in low-symmetry aromatic ring molecules without using any intricate quantum-optimal control procedure. This may open up an effective generation method of ring currents and current-induced magnetic fields in biomolecules such as amino acids having aromatic ring molecules for searching their interactions.