Oh no, I was too slow... This is Jensen's identity. It first appeared (in a slightly modified form) in: *Jensen, Sur une identité d'Abel et sur d'autres formules analogues, Acta Mathematica, vol 26 (1902) pp.307-318*.

There is the following paper that gives elementary proofs to Jensen's identity and some generalizations or related formulas: *Guo, On Jensen's and related combinatorial identities. Applicable Analysis and Discrete Mathematics
Vol. 5, No. 2 (2011), pp. 201-211*.

The elementary proof is very short, and consists just in a clever use of the Chu-Vandermonde convolution formula 
$$\sum_{k=0}^n \binom A k \binom B {n-k}=\binom {A+B} n\quad (1)$$ on the term $\binom {y-k\omega} k $ with $A=x+y+1$ and $A+B=y-k\omega $, followed by a straightforward change in the order of summation.  Then you get an equivalent identity  (just change of the names of the variables).

The proof of Chu's and Mohanty-Handa's multinomial generalization of Jensen's identity is similar, and involves a multinomial Chu-Vandermonde used iteratively.

Since (1) has combinatorial significance, it makes sense to rewrite Guo's computations avoiding changes of variables and negative binomials. I think that you get even a slightly more transparent (though, equivalent) way of presenting the proof. A key summation here is (please check my signs):
$$ \sum_{0\leq k\leq i\leq n} \binom {x+k\omega} k (-1)^{n-k} \binom {x+k\omega-k} {i-k} \binom {x+y-i} {n-i}. $$

Now it is clear that if you sum over $i$ first, you get the LHS of Jensen's identity, while you get the RHS summing over $k$. Moreover, both directions can be proved combinatorially (assuming at least that $x,y, and $\omega$ are integers).