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We report on a quantum critical behavior in the quasi-1D spin-1/2 zigzag frustrated chain antiferromagnet CaCoV_{2}O_{7}, 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 J_{1} and J_{2} of the zigzag spin chain.