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We establish the universal approximation capability of single-layer, single-head self- and cross-attention mechanisms with minimal attached structures. Our key insight is to interpret single-head attention as an input domain-partition mechanism that assigns distinct values to subregions. This allows us to engineer the attention weights such that this assignment imitates the target function. Building on this, we prove that a single self-attention layer, preceded by sum-of-linear transformations, is capable of approximating any continuous function on a compact domain under the $L_\infty$-norm. Furthermore, we extend this construction to approximate any Lebesgue integrable function under $L_p$-norm for $1\leq p <\infty$. Lastly, we also extend our techniques and show that, for the first time, single-head cross-attention achieves the same universal approximation guarantees.