For successful estimation, the usual network tomography algorithms crucially require i) end-to-end data generated using multicast probe packets, real or emulated, and ii) the network to be a tree rooted at a single sender with destinations at leaves. These requirements, consequently, limit their scope of application. In this paper, we address successfully a general problem, henceforth called generalized network tomography, wherein the objective is to estimate the link performance parameters for networks with arbitrary topologies using only end-to-end measurements of pure unicast probe packets. Mathematically, given a binary matrix $A,$ we propose a novel algorithm to uniquely estimate the distribution of $X,$ a vector of independent non-negative random variables, given only IID samples of the components of the random vector $Y = AX.$ This algorithm, in fact, does not even require any prior knowledge of the unknown distributions. The idea is to approximate the distribution of each component of $X$ using linear combinations of known exponential bases and estimate the unknown weights. These weights are obtained by solving a set of polynomial systems based on the moment generating function of the components of $Y.$ For unique identifiability, it is only required that every pair of columns of the matrix $A$ be linearly independent, a property that holds true for the routing matrices of all multicast tree networks. Matlab based simulations have been included to illustrate the potential of the proposed scheme.