We investigate how cosmic web structures affect galaxy quenching in the IllustrisTNG (TNG100) cosmological simulations by reconstructing the cosmic web within each snapshot using the DisPerSE framework. We measure the comoving distance from each galaxy with stellar mass
log
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⊙
)
≥
8
to the nearest node (d
node) and the nearest filament spine (d
fil) to study the dependence of both the median specific star formation rate (〈sSFR〉) and the median gas fraction (〈f
gas〉) on these distances. We find that the 〈sSFR〉 of galaxies is only dependent on the cosmic web environment at z < 2, with the dependence increasing with time. At z ≤ 0.5,
8
≤
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/
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<
9
galaxies are quenched at d
node ≲ 1 Mpc, and have significantly suppressed star formation at d
fil ≲ 1 Mpc, trends driven mostly by satellite galaxies. At z ≤ 1, in contrast to the monotonic drop in 〈sSFR〉 of
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galaxies with decreasing d
node and d
fil,
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≥
10
galaxies—both centrals and satellites—experience an upturn in 〈sSFR〉 at d
node ≲ 0.2 Mpc. Much of this cosmic web dependence of star formation activity can be explained by an evolution in 〈f
gas〉. Our results suggest that in the past ∼10 Gyr, low-mass satellites are quenched by rapid gas stripping in dense environments near nodes and gradual gas starvation in intermediate-density environments near filaments. At earlier times, cosmic web structures efficiently channeled cold gas into most galaxies. State-of-the-art ongoing spectroscopic surveys such as the Sloan Digital Sky Survey and DESI, as well as those planned with the Subaru Prime Focus Spectrograph, JWST, and Roman, are required to test our predictions against observations.