The production of K(S)(0) and Lambda hadrons is studied in pp collision data at root s = 0.9 and 7 TeV collected with the ATLAS detector at the LHC using a minimum-bias trigger. The observed distributions of transverse momentum, rapidity, and multiplicity are corrected to hadron level in a model-independent way within well-defined phase-space regions. The distribution of the production ratio of (Lambda) over bar to Lambda baryons is also measured. The results are compared with various Monte Carlo simulation models. Although most of these models agree with data to within 15% in the K(S)(0) distributions, substantial disagreements are found in the Lambda distributions of transverse momentum.