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 $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ Even though I rarely need exact sequences in my work, whenever I'm working with tensor products I find that simple manipulations like this tend to make life a lot easier.
(Interestingly, I don't know a textbook that mentions $(*)$ explicitly. Most linear algebra textbooks never even mention exact sequences, and most abstract algebra textbooks hide $(*)$ behind a general statement such as "every projective module is flat".)