This article proposes and studies a new three-parameter generalized model of the inverse Gompertz distribution, in the so-called Kumaraswamy inverse Gompertz distribution. The main advantage of the new model is that it has "an upside down bathtub-shaped curve hazard rate function" depending upon the shape parameters. Several of its statistical and mathematical properties including quantiles, median, mode, moments, probability weighted moment, entropy function, skewness and kurtosis are derived. Moreover, the reliability and hazard rate functions, mean time to failure, mean residual and inactive lifetimes are also concluded. The maximum likelihood approach is done here to estimate the new model parameters. A simulation study is conducted to examine the performance of the estimators of this model. Finally, the usefulness of the proposed distribution is illustrated with different engineering applications to complete, type-II right censored, and upper record data and it is found that this model is more flexible when it is compared to well-known models in the statistical literature.