We present the first ≥3.5 (conservative) or ≥5.8 σ (sum-of-lines significance) detection of two warm-hot intergalactic medium (WHIM) filaments at z > 0, which we find along the line of sight to the blazar Mrk 421. These systems are detected through highly ionized resonant metal absorption in high-quality Chandra ACIS and HRC Low Energy Transmission Grating (LETG) spectra of Mrk 421, obtained following our two Target of Opportunity requests during two outburst phases (F_{0.5-2 keV} = 40 and 60 mcrab; 1 crab = 2 × 10^-8 ergs s^-1 cm^-2). Columns of He-like oxygen and H-like nitrogen can be detected in the co-added LETG spectrum of Mrk 421 down to a sensitivity of N ≥ 8 × 10¹⁴ cm^-2 and N ≥ 10¹⁵ cm^-2, respectively, at a significance ≥3 σ. The two intervening WHIM systems that we detect have O VII and N VII columns of N = (1.0 ± 0.3) × 10¹⁵ cm^-2, N = (0.8 ± 0.4) × 10¹⁵ cm^-2 and N = (0.7 ± 0.3) × 10¹⁵ cm^-2, N = (1.4 ± 0.5) × 10¹⁵ cm^-2, respectively. We identify the closest of these two systems with an intervening WHIM filament at cz = 3300 ± 300 km s^-1. The second system, instead, at cz = 8090 ± 300 km s^-1, is identified with an intervening WHIM filament located ~13 Mpc from the blazar. The filament at cz = 3300 ± 300 lies 5 Mpc from a known H I Lyα system at cz = 3046 ± 12 km s^-1 (Shull and coworkers) whose 3 σ maximal H I kinetic temperature, as derived from the observed line FWHM, is T ≤ 1.2 × 10⁵ K. This temperature is inconsistent with the temperature measured for the X-ray filament, so if the systems are related, a multiphase WHIM is required. Combining UV and far-ultraviolet (FUV) upper limits on the H I Lyα and the O VI_2s→2p transitions with our measurements in the X-rays, we show that, for both filaments, equilibrium collisional ionization (with residual photoionization by both the diffuse UV and X-ray background and the beamed emission of Mrk 421 along our line of sight) provides acceptable solutions. These solutions define ranges of temperatures, metallicity ratios, and equivalent H column densities, which are in good agreement with the predictions of hydrodynamical simulations for the formation of large-scale structures in the universe. From the detected number of WHIM filaments along this line of sight we can estimate the number of O VII filaments per unit redshift with columns larger than 7 × 10¹⁴ cm^-2, d/dz(N ≥ 7 × 10¹⁴) = 67, consistent, within the large 1 σ errors, with the hydrodynamical simulation predictions of d/dz(N ≥ 7 × 10¹⁴) = 30. Finally, we measure a cosmological mass density of X-ray WHIM filaments Ω_b = 0.027 × 10, consistent with both model predictions and the estimated number of "missing" baryons at low redshift.