An example that might be useful in virtually all branches of mathematics: If $V$ and $W$ are vector spaces (over the same field $\mathbb{F}$) and if $U \subseteq V$ is a subspace, then the obvious exact sequence $$ 0 \longrightarrow U \longrightarrow V \longrightarrow V/U \longrightarrow 0 $$ turns into the non-trivial exact sequence $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ (I rarely need exact sequences in my work, but simple manipulations like this make quotients and subspaces of tensor products much easier to deal with.)
J. van Dobben de Bruyn
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