We report the first observation of the doubly Cabibbo-suppressed decays $D^+ → K^+ η^{(\prime)}$ using a 791 fb$^{-1}$ data sample collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. The ratio of the branching fractions of doubly Cabibbo-suppressed relative to singly Cabibbo-suppressed $D^+ → π^+ η^{(\prime)}$ decays are $\mathcal{B}(D^+ → K^+ η)/\mathcal{B}(D^+ → π^+ η)$ = (3.06 $±$ 0.43 $±$ 0.14)% and $\mathcal{B}(D^+ → K^+ η')/\mathcal{B}(D^+ → π^+ η')$ = (3.77 $±$ 0.39 $±$ 0.10)%. From these, we find that the relative final-state phase difference between the tree and annihilation amplitudes in $D^+$ decays, $δ_{TA}$, is (72 $±$ 9)$^∘$ or (288 $±$ 9)$^∘$. We also report the most precise measurements of CP asymmetries to date: $A_{CP}^{D^+ → π^+ η}$ = (+1.74 $±$ 1.13 $±$ 0.19)% and $A_{CP}^{D^+ → π^+ η'}$ = ($-$0.12 $±$ 1.12 $±$ 0.17)%. ; Comment: 6 pages, 2 figures, 3 tables. Published in PRL