  Motivated by the recent surge of transverse-field experiments on
quasi-one-dimensional antiferromagnets Sr(Ba)Co$_2$V$_2$O$_8$, we investigate
the quantum phase transition in a Heisenberg-Ising chain under a combination of
two in-plane inter-perpendicular transverse fields and a four-period
longitudinal field, where the in-plane transverse field is either uniform or
staggered. We show that the model can be unitary mapped to the one-dimensional
transverse-field Ising model (1DTFIM) when the $x$ and $y$ components of the
spin interaction and the four-period field are absent. When these two terms are
present, following both analytical and numerical efforts, we demonstrate that
the system undergoes a second-order quantum phase transition with increasing
transverse fields, where the critical exponents as well as the central charge
fall into the universality of 1DTFIM. Our results naturally identify the 1DTFIM
universality of 1D quantum phase transitions observed in the existed
experiments in Sr(Ba)Co$_2$V$_2$O$_8$ with transverse field applied along
either [100] or [110] direction. Upon varying the tuning parameters a critical
surface with 1DTFIM universality is determined and silhouetted to exhibit the
general presence of the universality in a much wider scope of models than
conventional understanding. Thus our results provide a broad guiding framework
to facilitate the experimental realization of 1DTFIM universality in real
materials.
