We provide an algorithm giving a 140/41 (<3.415)-approximation for Coflow Scheduling and a 4.36-approximation for Coflow Scheduling with release dates. This improves upon the best known 4- and respectively 5-approximations and addresses an open question posed by Agarwal, Rajakrishnan, Narayan, Agarwal, Shmoys, and Vahdat [Aga+18], Fukunaga [Fuk22], and others. We additionally show that in an asymptotic setting, the algorithm achieves a (2+ϵ)-approximation, which is essentially optimal under P≠NP. The improvements are achieved using a novel edge allocation scheme using iterated LP rounding together with a framework which enables establishing strong bounds for combinations of several edge allocation algorithms.