We report the first dijet transverse momentum asymmetry measurements from Au+Au and pp collisions at RHIC. The two highest-energy back-to-back jets reconstructed from fragments with transverse momenta above 2 GeV/c display a significantly higher momentum imbalance in heavy-ion collisions than in the pp reference. When reexamined with correlated soft particles included, we observe that these dijets then exhibit a unique new feature-momentum balance is restored to that observed in pp for a jet resolution parameter of R=0.4, while rebalancing is not attained with a smaller value of R=0.2.