We study minimal thinness in the half-space $H:=\{x=(\tilde{x},x_d):\tilde{x}\in {ℝ}^{d-1},x_d\>0\}$ for a large class of subordinate Brownian motions. We show that the same test for the minimal thinness of a subset of $H$ below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.