We introduce Dunkl version of Laplace-Runge-Lenz vector associated with a finite Coxeter group \(W\) acting geometrically in \(\mathbb R^N\) with multiplicity function \(g\). This vector commutes with Dunkl Laplacian with additional Coulomb potential \(\gamma/r\), and it generalises the usual Laplace-Runge-Lenz vector. We study resulting symmetry algebra \(R_{g, \gamma}(W)\) and show that it has Poincar\'e-Birkhoff-Witt property. In the absence of Coulomb potential this symmetry algebra is a subalgebra of the rational Cherednik algebra \(H_g(W)\supset R_{g,0}(W)\). We show that its central quotient is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra \(H_g^{so(N+1)}(W)\). This gives interpretation of the algebra \(H_g^{so(N+1)}(W)\) as the hidden symmetry algebra of Dunkl Laplacian. On the other hand by specialising \(R_{g, \gamma}(W)\) to \(g=0\) we recover a quotient of the universal enveloping algebra \(U(so(N+1))\) as the hidden symmetry algebra of Coulomb problem in \({\mathbb R}^N\).