One of the most prominent features of galaxy clusters is the presence of a dominant population of massive ellipticals in their cores. Stellar archaeology suggests that these gigantic beasts assembled most of their stars in the early Universe via intense starbursts. However, the role of dense environments and their detailed physical mechanisms in triggering starburst activities remain unknown. Here we report spatially-resolved Atacama Large Millimeter/submillimeter Array (ALMA) observations of the molecular gas, with a resolution of about 2.5 kiloparsecs, toward a forming galaxy cluster core with intense starburst galaxies at z = 2.51. In contrast to starburst galaxies in the field often associated with galaxy mergers or highly turbulent gaseous disks, our observations show that the two starbursts in the cluster exhibit dynamically cold (rotation-dominated) gas-rich disks. Their gas disks have extremely low velocity dispersion (σ ∼ 20 − 30 km s−1 ), three times lower than their field counterparts at similar redshifts. The high gas fraction and suppressed velocity dispersion yield gravitationally unstable gas disks, which enables highly efficient star formation. The suppressed velocity dispersion, likely induced by accretion of co-rotating and co-planar cold gas, might serve as an essential avenue to trigger starbursts in forming galaxy clusters at high redshift.