We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a well-known notion of graph separators, we show that an embedding with a smaller edge-congestion can be obtained if the guest graph has a smaller separator, and if the host grid has a higher but constant dimension. Specifically, we prove that any graph with N nodes, maximum node degree Δ , and with a node-separator of size s , where s is a function such that s ( n ) = O ( n α ) with 0 ? α < 1 , can be embedded into a grid of a fixed dimension d ? 2 with at least N nodes, with an edge-congestion of O ( Δ ) if d 1 / ( 1 - α ) , O ( Δ log N ) if d = 1 / ( 1 - α ) , and O ( Δ N α - 1 + 1 d ) if d < 1 / ( 1 - α ) . This edge-congestion achieves constant ratio approximation if d 1 / ( 1 - α ) , and matches an existential lower bound within a constant factor if d ? 1 / ( 1 - α ) . Our result implies that if the guest graph has an excluded minor of a fixed size, such as a planar graph, then we can obtain an edge-congestion of O ( Δ log N ) for d = 2 and O ( Δ ) for any fixed d ? 3 . Moreover, if the guest graph has a fixed treewidth, such as a tree, an outerplanar graph, and a series-parallel graph, then we can obtain an edge-congestion of O ( Δ ) for any fixed d ? 2 . To design our embedding algorithm, we introduce edge-separators bounding extension, such that in partitioning a graph into isolated nodes using edge-separators recursively, the number of outgoing edges from a subgraph to be partitioned in a recursive step is bounded. We present an algorithm to construct an edge-separator with extension of O ( Δ n α ) from a node-separator of size O ( n α ) .