  The ATLAS and CMS collaborations have observed independently at the Large
Hadron Collider (LHC) a new Higgs-like particle with a mass $M_h \sim$ 125 GeV
and properties similar to that predicted by the Standard Model (SM). Although
the measurements indicate that this Higgs-like boson is compatible with the SM
hypothesis, however due to large uncertainties in some of the Higgs detection
channels, one still has the possibility of testing this object as being a
candidate for some Beyond the SM (BSM) physics scenarios, for example, the
Minimal Supersymmetric Standard Model (MSSM), in the CP-conserving version
(CPC-MSSM). In this paper, we evaluate the modifications of these CPC-MSSM
results when CP-violating (CPV) phases are turned on explicitly, leading to the
CP-violating MSSM (CPV-MSSM). We investigate the role of the CPV phases in
(some of) the soft Supersymmetry (SUSY) terms on both the mass of the lightest
Higgs boson $h_1$, and the rates for the processes $gg \rightarrow h_1
\rightarrow \gamma \gamma$, $gg \rightarrow h_1 \rightarrow ZZ^*\rightarrow
4l$, $gg \rightarrow h_1 \rightarrow WW^*\rightarrow l \nu l \nu$, $pp
\rightarrow V h_1 \rightarrow V b\bar b$ and $pp \rightarrow V h_1 \rightarrow
V \tau^+\tau^-$, ($V \equiv W^\pm, Z$) at the LHC, considering the impact of
the flavor constraints as well as the constraints coming from the Electric
Dipole Moment (EDM) measurements. We find that the imaginary part of the top
and bottom Yukawa couplings can take very small but non-zero values even after
satisfying the recent updates from LHC. Our study shows that the CPV-MSSM
provides an equally potential solution (like its CP-conserving (CPC)
counterpart) to the recent LHC Higgs data. Improvement in different Higgs
coupling measurements is necessary in order to test the possibility of probing
the small dependence on these CPV phases in the Higgs sector of the MSSM.
