We multiply two $n \times n$ matrices $S,T$ over semirings in the Congested Clique model, where $n$ fully connected nodes communicate synchronously using $O(\log n)$-bit messages, within $O(nz(S)^{1/3} nz(T)^{1/3}/n + 1)$ rounds of communication, where $nz(A)$ denotes the number of non-zero elements in a matrix $A$. By leveraging the sparsity of the input matrices, our algorithm greatly reduces communication compared with general algorithms [Censor-Hillel et al., PODC 2015], improving upon the state-of-the-art for matrices with $o(n^2)$ non-zero elements. Our algorithm exhibits the additional strength of surpassing previous solutions also when only one matrix is sparse. This allows efficiently raising a sparse matrix to a power greater than 2. As applications, we speed up 4-cycle counting and APSP in sparse graphs. Our algorithmic contribution is a new \emph{deterministic} method of restructuring the input matrices in a sparsity-aware manner, which assigns each node with element-wise multiplication tasks that are not necessarily consecutive but are balanced, yielding communication-efficient multiplication. Moreover, this new deterministic method for restructuring matrices may be used to restructure the adjacency matrix of input graphs, enabling faster solutions for graph related problems. As an example, we present a new deterministic algorithm which solves the triangle listing problem in $O(m/n^{5/3} + 1)$ rounds, a complexity that was previously obtained by a \emph{randomized} algorithm [Pandurangan et al., SPAA 2018] and matches the lower bound of $\tildeΩ(n^{1/3})$ when $m=n^2$ of [Izumi and Le Gall, PODC 2017, Pandurangan et al., SPAA 2018]. Our triangle listing algorithm implies triangle counting with the same complexity of $O(m/n^{5/3} + 1)$ rounds, which is a \emph{cubic} improvement over the previous $O(m^2/n^3)$-round algorithm [Dolev et al., DISC 2012].