Quantum spin liquids (QSLs) as novel phases of matter with long-range entanglement and deconfined quantum critical points (DQCPs) as descriptions for unconventional phase transitions between two ordered states beyond the standard paradigm, such as the transition between antiferromagnetic (AFM) and valence-bond solid (VBS) phases, are two representative emerging phenomena. These implications for understanding correlated materials and developing theoretical frameworks for many-body physics are of crucial importance. Here, we show that a gapless QSL can naturally emerge from a DQCP. Via large-scale tensor network simulations of a square-lattice spin-$1/2$ frustrated Heisenberg model, both QSL-state and DQCP-type AFM-VBS transitions are observed. By tuning the coupling constants, the AFM-VBS transition vanishes, and instead, a gapless QSL phase gradually develops in between. Remarkably, along the phase boundaries of AFM-QSL and QSL-VBS transitions, we always observe the same correlation-length exponents, $\nu \approx 1.0$, which is intrinsically different from the one of the DQCP-type transition, indicating new types of universality classes. Our results explicitly demonstrate a new scenario for understanding the emergence of gapless QSL from an underlying DQCP. The discovered QSL phase survives in a large region of tuning parameters, and we expect its experimental realization in solid-state materials or quantum simulators.