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Several rapid parameter estimation methods have recently been advanced to deal with the computational challenges of the problem of Bayesian inference of the properties of compact binary sources detected in the upcoming science runs of the terrestrial network of gravitational wave detectors. Some of these methods are well-optimized to reconstruct gravitational wave signals in nearly real-time necessary for multi-messenger astronomy. In this context, this work presents a new, computationally efficient algorithm for fast evaluation of the likelihood function using a combination of numerical linear algebra and mesh-free interpolation methods. The proposed method can rapidly evaluate the likelihood function at any arbitrary point of the sample space at a negligible loss of accuracy and is an alternative to the grid-based parameter estimation schemes. We obtain posterior samples over model parameters for a canonical binary neutron star system by interfacing our fast likelihood evaluation method with the nested sampling algorithm. The marginalized posterior distributions obtained from these samples are statistically identical to those obtained by brute force calculations. We find that such Bayesian posteriors can be determined within a few minutes of detecting such transient compact binary sources, thereby improving the chances of their prompt follow-up observations with telescopes at different wavelengths. It may be possible to apply the blueprint of the meshfree technique presented in this study to Bayesian inference problems in other domains.