This paper reports new exclusive cross sections for ep→e′π+π−p′ using the CLAS detector at Jefferson Laboratory. These results are presented for the first time at photon virtualities 2.0 GeV^2 < Q^2 < 5.0 GeV^2 in the center-of-mass energy range 1.4 GeV < W < 2.0 GeV, which covers a large part of the nucleon resonance region. Using a model developed for the phenomenological analysis of electroproduction data, we see strong indications that the relative contributions from the resonant cross sections at W < 1.74 GeV increase with Q2. These data considerably extend the kinematic reach of previous measurements. Exclusive ep→e′π+π−p′ cross section measurements are of particular importance for the extraction of resonance electrocouplings in the mass range above 1.6 GeV.