We release new measurements of the cosmic CO and [CII] line backgrounds using a tomographic intensity mapping approach. These mean line intensities represent the integrated background emission from all sources in the cosmic web, including galaxies too faint to detect individually, and potentially diffuse gas extending beyond galaxies. This work builds upon a companion study by Chiang et al. (2025, arXiv:2504.05384), which measured the redshift evolution of the cosmic infrared background (CIB) through spatial cross-correlations between diffuse intensity maps and reference galaxies over redshifts 0 < z < 4. Specifically, that study combines 11 maps from Planck, Herschel, and IRAS with over 3 million spectroscopic galaxies from SDSS, BOSS, and eBOSS, producing a tomographic data vector of CIB–galaxy correlations across a wide frequency-redshift domain. The companion paper focused on interpreting and extracting information from the CIB continuum, leading to a census of cosmic dust and star formation history. Here, we extend and reanalyze the same tomographic cross-correlation data to isolate line contributions from CO and [CII], key spectral features in the far-infrared to millimeter wave for galaxy studies and intensity mapping. These lines are identified statistically through their frequency- and redshift-dependent imprint in the CIB-galaxy cross-correlation amplitudes. Our continuum-plus-line fitting yields the first detections of the cosmic CO background at 7σ significance and [CII] at 3σ over 0 < z < 4. The [CII] signal corresponds to the 158-micron fine-structure line, while the CO background includes nine rotational transitions from J = 1–0 to J = 9–8, starting at 115.27 GHz with uniform frequency spacing. The released data file provides empirically measured sky monopole brightness temperatures of [CII] and nine CO lines in microkelvin (μK), tabulated as functions of redshift over 0 < z < 12 and observed frequency in GHz. Best-fit values correspond to posterior medians, with 68% credible intervals also provided, representing the first constraints on the cosmic mean CO and [CII] intensities. This data release provides a key empirical baseline for future CO and [CII] line-intensity mapping experiments to anchor expected signal strengths, facilitating survey designs and forecasts to unlock potential astrophysical and cosmological applications. It also serves as a reference for comparison with galaxy surveys in studies of star formation, molecular gas, and large-scale structure using these lines. Version 2 note: This release incorporates the corrected Herschel SPIRE filter response functions for extended sources as described in the SPIRE Observers’ Manual, replacing incorrect profiles previously used from the SVO database. This update is propagated from the companion work (Chiang et al. 2025, arXiv:2504.05384), leading to only minor changes in the recovered CO and [CII] amplitudes.