Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial hight homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on the universal cover of $Y$ via
$$([\sigma],y)\mapsto [\sigma \circ f]y$$
If g is another continuous map whose lift to the universal covers is equivariant under these actions, is g homotopic to f?
Your aspherical spaces are just $K(\pi_1(X),1)$. It is an easy fact that homotopy classes of maps between Eilenberg-Maclane spaces are in bijective correspondence with maps between their respective homotopy group. Since the deck transformation group of the universal cover is naturally isomorphic to the fundamental group, your statement should follow if I understand it correctly.
The special case you're interested in can be found here: https://www.math.cornell.edu/~hatcher/AT/ATch1.pdf on page 90 as Proposition 1B.9.