One important area of statistical theory and its applications to bivariate data modeling is the construction of families of bivariate distributions with specified marginals. This motivates the proposal of a bivariate distribution employing the Farlie-Gumbel-Morgenstern (FGM) copula and Epanechnikov exponential (EP-EX) marginal distribution, denoted by EP-EX-FGM. The EP-EX distribution is a complementing distribution, not a rival, to the exponential (EX) distribution. Its simple function shape and dependence on a single scale parameter make it an ideal choice for marginals in the suggested new bivariate distribution. The statistical properties of the EP-EX-FGM model are examined, including product moments, coefficient of correlation between the internal variables, moment generating function, conditional distribution, concomitants of order statistics (OSs), mean residual life function, and vitality function. In addition, we calculated reliability and information measures including the hazard function, reversed hazard function, positive quadrant dependence feature, bivariate extropy, bivariate weighted extropy, and bivariate cumulative residual extropy. Estimating model parameters is accomplished by utilizing maximum likelihood, asymptotic confidence intervals, and Bayesian approaches. Finally, the advantage of EP-EX-FGM over the bivariate Weibull FGM distribution, bivariate EX-FGM distribution, and bivariate generalized EX-FGM distribution is illustrated using actual data sets.