We study the unitary dynamics of randomly or quasiperiodically driven tilted Bose-Hubbard (tBH) model in one dimension deep inside its Mott phase starting from a Z(2) symmetry-broken state. The randomness is implemented via a telegraph noise protocol in the drive period while the quasiperiodic drive is chosen to correspond to a Thue-Morse sequence. The periodically driven tBH model (with a square pulse protocol characterized by a time period T) is known to exhibit transitions from dynamical regimes with long-time coherent oscillations to those with rapid thermalization. Here we show that starting from a regime where the periodic drive leads to rapid thermalization, a random drive, which consists of a random sequence of square pulses with period T + alpha dT, where alpha = +/- 1 is a random number and dT is the amplitude of the noise, restores long-time coherent oscillations for special values of dT, A similar phenomenon can be seen for a quasiperiodic drive following a Thue-Morse sequence where such coherent behavior is shown to occur for a larger number of points in the (T, dT) plane due to the additional structure of the drive protocol. We chart out the dynamics of the system in the presence of such aperiodic drives, provide a qualitative analytical understanding of this phenomenon, point out the role of quantum scars behind it, and discuss experiments which can test our theory.