I know that it holds even if $X$ has infinite dimension, but I am looking for a specific argument in the finite-dimensional case.
The classical (and short) argument is the one in
A. Hatcher: Algebraic topology, Cambridge University Press (ISBN 0-521-79540-0/pbk). xii, 544 p. (2002). ZBL1044.55001,
see Proposition A.4 p. 523. I do not think there is any significant simplification taking $X$ finite-dimensional.