The dynamics of surface-attached polymers play a key role in the operation of a number of biological sensors, yet its current understanding is rather limited. Here we use computer simulations to study the dynamics of a reaction between the free end of a polymer chain and a surface, to which its other end has been attached. We consider two limiting cases, the diffusion-control led limit, where the reaction is accomplished whenever the free chain end diffuses to within a specified distance from the Surface, and the reaction-controlled limit, where slow, intrinsic reaction kinetics rather than diffusion of the chain is rate limiting. In the diffusion-controlled limit, we find that the overall rate scales as N-b, where N is the number of monomers in the chain and b approximate to 2.2 for excluded Volume chains, This value of the scaling exponent b is close to that derived from a simple approximate theory treating the dynamics of the chain end relative to the Surface its one-dimensional diffusion in an effective potential. In the reaction-control led limit, the value of the scaling exponent b is close to 1. We compare our findings with those for the related (and better Studied) problem of end-to-end reactions within all unconstrained polymer chain and discuss their implications for electrochemical DNA sensors.