There is a precise formulation of this fact in Toen's master course on stacks, although a proof is not given:
First, we define the complementary subfunctor of a closed immersion $Y\hookrightarrow X$ to be a particular subfunctor $X - Y\hookrightarrow X$, which is then proven to itself be an open immersion (see example 4 section 4 of Cours 4). It is then noted (without proof) in proposition 1 of section 1 of Cours 4.5 that for every open immersion $U\hookrightarrow X$, there exists a closed immersion $V\hookrightarrow X$ such that $U\cong X - V$, which is, if I remember correctly, not unique (although I believe there is a universal (either initial or terminal) such closed immersion (this universal object is analogous to the "reduced induced" closed subscheme structure on a closed subset of a scheme).