Two-particle correlations in relative azimuthal angle (Δϕ) and pseudorapidity (Δη) are measured in √sNN=5.02  TeV p+Pb collisions using the ATLAS detector at the LHC. The measurements are performed using approximately 1  μb−1 of data as a function of transverse momentum (pT) and the transverse energy (ΣEPbT) summed over 3.1<η<4.9 in the direction of the Pb beam. The correlation function, constructed from charged particles, exhibits a long-range (2<|Δη|<5) “near-side” (Δϕ∼0) correlation that grows rapidly with increasing ΣEPbT. A long-range “away-side” (Δϕ∼π) correlation, obtained by subtracting the expected contributions from recoiling dijets and other sources estimated using events with small ΣEPbT, is found to match the near-side correlation in magnitude, shape (in Δη and Δϕ) and ΣEPbT dependence. The resultant Δϕ correlation is approximately symmetric about π/2, and is consistent with a dominant cos2Δϕ modulation for all ΣEPbT ranges and particle pT.