Models of galaxy formation predict that gas accretion from the cosmic web is a primary driver of star formation over cosmic history. Except in very dense environments where galaxy mergers are also important, model galaxies feed from cold streams of gas from the web that penetrate their dark matter halos. Although these predictions are unambiguous, the observational support has been indirect so far. Here, we report spectroscopic evidence for this process in extremely metal-poor galaxies (XMPs) of the local universe, taking the form of localized starbursts associated with gas having low metallicity. Detailed abundance analyses based on Gran Telescopio Canarias optical spectra of 10 XMPs show that the galaxy hosts have metallicities around 60% solar, on average, while the large star-forming regions that dominate their integrated light have low metallicities of some 6% solar. Because gas mixes azimuthally in a rotation timescale (a few hundred Myr), the observed metallicity inhomogeneities are only possible if the metal-poor gas fell onto the disk recently. We analyze several possibilities for the origin of the metal-poor gas, favoring the metal-poor gas infall predicted by numerical models. If this interpretation is correct, XMPs trace the cosmic web gas in their surroundings, making them probes to examine its properties.