A longstanding open problem in Algorithmic Mechanism Design is to design computationally-efficient truthful mechanisms for (approximately) maximizing welfare in combinatorial auctions with submodular bidders. The first such mechanism was obtained by Dobzinski, Nisan, and Schapira [STOC'06] who gave an O(log2m)-Approximation where m is number of items. This problem has been studied extensively since, culminating in an O(√log m)-Approximation mechanism by Dobzinski~[STOC'16]. We present a computationally-efficient truthful mechanism with approximation ratio that improves upon the state-of-The-Art by an exponential factor. In particular, our mechanism achieves an O((log logm)3)-Approximation in expectation, uses only O(n) demand queries, and has universal truthfulness guarantee. This settles an open question of Dobzinski on whether Θ(√log m) is the best approximation ratio in this setting in negative.