This is the publisher's version, also available electronically from http://journals.aps.org/prd/abstract/10.1103/PhysRevD.88.052001.Measurements are presented of the production of primary K(0)(S) and Λ particles in proton-proton collisions at s√=7 TeV in the region transverse to the leading charged-particle jet in each event. The average multiplicity and average scalar transverse momentum sum of K(0)(S) and Λ particles measured at pseudorapidities |η|<2 rise with increasing charged-particle jet p(T) in the range 1–10 GeV/c and saturate in the region 10–50 GeV/c. The rise and saturation of the strange-particle yields and transverse momentum sums in the underlying event are similar to those observed for inclusive charged particles, which confirms the impact-parameter picture of multiple parton interactions. The results are compared to recent tunes of the pythia Monte Carlo event generator. The pythia simulations underestimate the data by 15%–30% for K(0)(S) mesons and by about 50% for Λ baryons, a deficit similar to that observed for the inclusive strange-particle production in non-single-diffractive proton-proton collisions. The constant strange- to charged-particle activity ratios with respect to the leading jet p(T) and similar trends for mesons and baryons indicate that the multiparton-interaction dynamics is decoupled from parton hadronization, which occurs at a later stage