This paper proposes a computationally efficient approach for estimating high dimensional Tobit models with censored outcomes, based on ordinary least squares (OLS) and Lasso regularization. Under a subgaussian and linearity condition on the covariates, which is mild in ultrahigh-dimensional settings, we derive non-asymptotic error bounds for both the Tobit OLS and Lasso estimators. When the subgaussian condition does not hold, we apply a Gaussianization technique to the covariates and show that the resulting Lasso estimator attains a comparable error bound, supporting the applicability of our method across diverse data distributions. Extensive numerical studies confirm the computational efficiency and estimation accuracy of the proposed approach. The empirical application shows that our approach consistently achieves a parsimonious model with rapid computation.