Measurements of specific heat and electrical resistivity in magnetic fields up to 9 T along [001] and temperatures down to 50 mK of Sn-substituted CeCoIn5 are reported. The maximal -ln(T) divergence of the specific heat at the upper critical field Hc2 down to the lowest temperature characteristic of non-Fermi-liquid systems at the quantum critical point (QCP), the universal scaling of the Sommerfeld coefficient, and agreement of the data with spin-fluctuation theory provide strong evidence for quantum criticality at Hc2 for all x< or =0.12 in CeCoIn5-xSnx. These results indicate the "accidental" coincidence of the QCP located near Hc2 in pure CeCoIn5, in actuality, constitute a novel quantum critical point associated with unconventional superconductivity.