Method of recovering oil from a subterranean oil-bearing reservoir by using an injection fluid containing a viscosifying polymer in a low salinity water. The low salinity water has a total dissolved solids (TDS) content of 15,000 ppmv or less, and the ratio of the multivalent cation content of the low salinity water to the multivalent cation content of the connate water of the reservoir is less than 1. The viscosity of the injection fluid is in the range of 3 to 200 cP, and the reservoir is penetrated by one or more injection wells and by one or more production wells. The method includes injecting the injection fluid into at least one of the injection wells in a slug size in the range of 0.4 to 1.5 pore volume (PV).