We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of $ \tilde Ω(\sqrt{n} + D) $ rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where $ n $ is the number of nodes in the network and $ D $ is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs $ \tilde O(n^{2/3} D^{1/3} + n^{5/6}) $ rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing $ \tilde O (\sqrt{n D}) $ rounds and the second performing $ \tilde O (\sqrt{n} D^{1/4} + n^{3/5} + D) $ rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a $ (1 + ε) $-approximation $ \tilde O ((\sqrt{n} D^{1/4} + D) / ε) $-round algorithm for directed SSSP and a new work/depth trade-off for exact SSSP on directed graphs in the PRAM model.