We study the effects from light sfermions on the lightest Higgs boson production and decay at the Large Hadron Collider (LHC) within the Minimal Supersymmetric Standard Model (MSSM). We find that the scenario with light coloured sfermions - stops or sbottoms - has the potential to explain a non-universal alteration, as hinted by LHC data, of the gluon-gluon Fusion (mu(ggF)) with respect to the Vector Boson Fusion (mu(VBF)) event rates and, in particular, can predict mu(VBF)/mu(ggF) > 1 for all Higgs boson decay channels in large areas of the parameter space. We also find that the scenario with a light stop/sbottom can be complemented by the scenario in which the total Higgs width, Gamma(tot), is reduced due to a suppressed Yukawa coupling Y-b. In this case, the reduction of the Higgs production rates in the ggF process which occurs in the maximal mixing scenario is compensated by the reduction of the H -> b (b) over bar partial decay width, the largest component of Gamma(tot). Furthermore, we highlight the fact that, in the light stop/sbottom scenario, event rates with the Higgs boson decaying to a b (b) over bar final state are predicted to be essentially below unity, especially in case of g g F, which is doubly suppressed, at production, due to the negative interference from stop/sbottom loops, as well at decay level, due to the Y-b suppression. Therefore, during the future LHC runs, the measurement of h -> b (b) over bar final states is a matter of special importance, which will offer additional handles to pin down the possible MSSM structure of the Higgs sector. Amongst all viable MSSM configurations that we study (including revisiting a light stau solution), we emphasise most the scenario with a light stop, as the latter is also motivated by Dark Matter and Electro-Weak baryogenesis. We also perform fits of the MSSM against the LHC data for all scenarios which we introduce, emphasising the fact that in most cases these are better than for the SM.