Consider the regular (n-1)-simplex $x_1+x_2+\cdots+x_n=k$ and $x_i\geq 0$. The collection of hyperplanes $x_i=p$ as well as the hyperplanes $x_i-x_j=q$ where $1\le i,j\le n$, $p.q\in \mathbb Z$, partition $\mathbb R^n$ into simplices with disjoint interiors (this is the Coxeter arrangement of type $A_{n}$, the small simplices are called alcoves). When restricted to our $(n-1)$-simplex we get a triangulation into $(n-1)$ dimensional simplices.
These $(n-1)$ dimensional simplices also partition the positive orthant $\mathbb R_+^n$ into several cones with disjoint interior, by taking their cone from the origin. You can show that each such cone is unimodular, in the sense that its generating rays form a basis of your lattice. And this implies your claim.
Lam and Potnikov wrote in Alcoved polytopes I several equivalent combinatorial ways to get these unimodular triangulations.