We propose a new imaging technique for interferometry using sparse modeling, utilizing two regularization terms: the $\ell_1$-norm and a new function named total squared variation (TSV) of the brightness distribution. First, we demonstrate that our technique may achieve a superresolution of $\sim$30% compared with the traditional CLEAN beam size using synthetic observations of two point sources. Second, we present simulated observations of three physically motivated static models of Sgr A$^\ast$ with the Event Horizon Telescope (EHT) to show the performance of proposed techniques in greater detail. Remarkably, in both the image and gradient domains, the optimal beam size minimizing root-mean-squared errors is $\le$ 10% of the traditional CLEAN beam size for $\ell_1$+ TSV regularization, and non-convolved reconstructed images have smaller errors than beam-convolved reconstructed images. This indicates that TSV is well matched to the expected physical properties of the astronomical images and the traditional post-processing technique of Gaussian convolution in interferometric imaging may not be required. We also propose a feature-extraction method to detect circular features from the image of a black hole shadow and use it to evaluate the performance of the image reconstruction. With this method and reconstructed images, the EHT can constrain the radius of the black hole shadow with an accuracy of $\sim$10% - 20% in present simulations for Sgr A$^\ast$, suggesting that the EHT would be able to provide useful independent measurements of the mass of the supermassive black holes in Sgr A$^\ast$ and also another primary target, M87.