If $X$ and $Y$ are finite type nilpotent spaces, then $X$ and $Y$ are weakly equivalent if and only if their cochain complexes are quasi-isomorphic as $E_\infty$-algebras. Moreover, assuming $X$ and $Y$ are also CW-complexes, two maps $f,g:X\to Y$ are homotopic if and only if they induce homotopic maps of $E_\infty$-algebras at the level of cochain complexes. In particular, if a map between such spaces $f:X\to Y$ induces a quasi-isomorphism in singular cohomology, then it is an homotopy equivalence. All these results are consequences of more precise theorems from this paper:

M. Mandell, Cochains and homotopy type, Publ. Math. IHES 103 (2006), 213, 246.

If $X$ and $Y$ are finite type nilpotent spaces, then $X$ and $Y$ are weakly equivalent if and only if their cochain complexes are quasi-isomorphic as $E_\infty$-algebras. Moreover, assuming $X$ and $Y$ are also CW-complexes, two maps $f,g:X\to Y$ are homotopic if and only if they induce homotopic maps of $E_\infty$-algebras at the level of cochain complexes. In particular, if a map between such spaces $f:X\to Y$ induces a quasi-isomorphism in singular cohomology, then it is an homotopy equivalence. All these results are consequences of more precise theorems from this paper:

M. Mandell, Cochains and homotopy type, Publ. Math. IHES 103 (2006), 213-246.