We report on a quantum critical behavior in the quasi-1D spin-1/2 zigzag frustrated chain antiferromagnet CaCoV₂O₇, induced by an applied magnetic field. Below T_N = 3.3 K our zero-field neutron diffraction studies revealed the up-up-down-down spin structure, stabilized by an order-by-disorder phenomenon. At base temperature, the magnetic order is suppressed by an applied magnetic field (B), inducing a transition into a quantum paramagnetic state at B_c = 3 T, as revealed by both neutron diffraction and ESR data. The transition exhibits an unusually sharp phase boundary with the critical exponent φ = 0.164(3) ≈ 1/6, in contrast to the earlier experimental observations for uniform spin-1/2 chain systems. Such a sharp QPT is anticipated due to a rare combination of spin-orbit coupling and competing NN and NNN exchange interactions J₁ and J₂ of the zigzag spin chain.