We report the first observation of the doubly Cabibbo-suppressed decays D+→K+η(′) using a 791fb-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(′) decays are B(D +→K+η)/B(D+→π+η )=(3.06±0.43±0.14)% and B(D+→K+η ′)/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, δTA, is (72±9)°or (288±9)°. We also report the most precise measurements of CP asymmetries to date: ACPD+→π +η=(+1.74±1.13±0.19)% and ACPD+→ π+η′=(-0.12±1.12±0.17)%. © 2011 American Physical Society.