[This should probably be a comment, since it is so short. Nevertheless, it is an answer to the question.]
The result you ask for is a consequence of proposition 2.2.15 in Sam Isaacson's Ph.D. thesis (Harvard, 2009). That proposition is stated for combinatorial symmetric monoidal (closed) model categories, and not just for simplicial sets.
I have no idea who first proved or published the above result.