One of the most prominent features of galaxy clusters is the presence of adominant population of massive ellipticals in their cores. Stellar archaeologysuggests that these gigantic beasts assembled most of their stars in the earlyUniverse via starbursts. However, the role of dense environments and theirdetailed physical mechanisms in triggering starburst activities remain unknown.Here we report spatially resolved Atacama Large Millimeter/submillimeter Array(ALMA) observations of the CO $J= 3-2$ emission line, with a resolution ofabout 2.5 kiloparsecs, toward a forming galaxy cluster core with starburstgalaxies at $z=2.51$. In contrast to starburst galaxies in the field oftenassociated with galaxy mergers or highly turbulent gaseous disks, ourobservations show that the two starbursts in the cluster exhibit dynamicallycold (rotation-dominated) gas-rich disks. Their gas disks have extremely lowvelocity dispersion ($\sigma_{\mathrm{0}} \sim 20-30$ km s$^{-1}$), which isthree times lower than their field counterparts at similar redshifts. The highgas fraction and suppressed velocity dispersion yield gravitationally unstablegas disks, which enables highly efficient star formation. The suppressedvelocity dispersion, likely induced by the accretion of corotating and coplanarcold gas, might serve as an essential avenue to trigger starbursts in massivehalos at high redshifts.