We provide a heuristic argument for disorder rounding of a first order quantum phase transition into a continuous quantum phase transition. This is illustrated by renormalization group analysis of the N-color Ashkin-Teller chain, the q- state Potts chain, the biquadratic spin one chain and the O(N) vector model with cubic anisotropy. In most of the cases, the results of renormalization group analysis are in excellent agreement with our heuristic argument. In a certain range of parameter space of the Ashkin-Teller model renormalization group calculations break down and indicate lack of universality. This may imply the persistence of the first order quantum phase transition and a possible modification of Aizenman-Wehr theorem for the quantum phase transitions.