We present the first observations of foreground Lyα forest absorption from high-redshift galaxies, targeting 24 star-forming galaxies (SFGs) with z ∼ 2.3-2.8 within a 5′ × 14′ region of the COSMOS field. The transverse sightline separation is ∼2 h-1 Mpc comoving, allowing us to create a tomographic reconstruction of the three-dimensional (3D) Lyα forest absorption field over the redshift range 2.20 ≤ z ≤ 2.45. The resulting map covers 6 h-1 Mpc × 14 h-1 Mpc in the transverse plane and 230 h-1 Mpc along the line of sight with a spatial resolution of ≈3.5 h-1 Mpc, and is the first high-fidelity map of a large-scale structure on ∼ Mpc scales at z > 2. Our map reveals significant structures with ≳10 h-1 Mpc extent, including several spanning the entire transverse breadth, providing qualitative evidence for the filamentary structures predicted to exist in the high-redshift cosmic web. Simulated reconstructions with the same sightline sampling, spectral resolution, and signal-to-noise ratio recover the salient structures present in the underlying 3D absorption fields. Using data from other surveys, we identified 18 galaxies with known redshifts coeval with our map volume, enabling a direct comparison with our tomographic map. This shows that galaxies preferentially occupy high-density regions, in qualitative agreement with the same comparison applied to simulations. Our results establish the feasibility of the CLAMATO survey, which aims to obtain Lyα forest spectra for ∼1000 SFGs over ∼1deg2 of the COSMOS field, in order to map out the intergalactic medium large-scale structure at 〈z〉 ∼ 2.3 over a large volume (100 h-1 Mpc)3.