Esscher transformed Laplace distribution is a new class of asymmetric heavy tailed distribution. In this article, we generalize the Esscher transformed Laplace distribution using the quadratic rank transmutation map to develop transmuted Esscher transformed Laplace distribution. We derived the probability density function of transmuted Esscher transformed Laplace distribution and its various properties were studied. The maximum likelihood estimation procedure is employed to estimate the parameters of the proposed distribution and an algorithm in R package is developed to carry out the estimation. Simulation studies for various choices of parameter values were performed to validate the algorithm. Finally, we fitted the transmuted Esscher transformed Laplace, Esscher transformed Laplace and Gaussian distributions to microarray gene expression dataset and compared them.