Integrated circuit (IC) camouflaging is a layout-level technique that thwarts reverse engineering attacks on ICs by introducing camouflaged cells that look alike, but can implement one of many possible Boolean functions. Existing camouflaging techniques have been broken by a recent decamouflaging attack, which uses Boolean satisfiability (SAT) techniques to compute specialized discriminating input patterns that prune the functionality search space quickly. This paper presents CamoPerturb, a countermeasure to thwart the decamouflaging attack by integrating logic perturbation with IC camouflaging. CamoPerturb, contrary to all the existing camouflaging schemes, perturbs the functionality of the given design minimally, i.e., adds/removes one minterm, rather than camouflaging the design. A separate camouflaged block CamoFix restores the perturbed minterm, recovering the functionality of the design. The perturbed minterm is the designer's secret and is incorporated into CamoFix using camouflaged cells. CamoPerturb renders the decamouflaging attack effort exponentially harder in the number of camouflaged gates while its overhead grows linearly. The paper presents formal proofs for the security of CamoPerturb along with experimental results.