In order to study further the long-range correlations ("ridge") observed recently in p + Pb collisions at root s(NN) = 5.02 TeV, the second-order azimuthal anisotropy parameter of charged particles, v(2), has been measured with the cumulant method using the ATLAS detector at the LHC. In a data sample corresponding to an integrated luminosity of approximately 1 mu b(-1), the parameter v(2) has been obtained using two- and four-particle cumulants over the pseudorapidity range vertical bar eta vertical bar < 2.5. The results are presented as a function of transverse momentum and the event activity, defined in terms of the transverse energy summed over 3.1 < eta < 4.9 in the direction of the Pb beam. They show features characteristic of collective anisotropic flow, similar to that observed in Pb + Pb collisions. A comparison is made to results obtained using two-particle correlation methods, and to predictions from hydrodynamic models of p + Pb collisions. Despite the small transverse spatial extent of the p + Pb collision system, the large magnitude of v(2) and its similarity to hydrodynamic predictions provide additional evidence for the importance of final-state effects in p + Pb reactions.