This paper focuses on the consensus averaging problem on graphs under general noisy channels. We study a particular class of distributed consensus algorithms based on damped updates, and using the ordinary differential equation method, we prove that the updates converge almost surely to exact consensus for finite variance noise. Our analysis applies to various types of stochastic disturbances, including errors in parameters, transmission noise, and quantization noise. Under a suitable stability condition, we prove that the error is asymptotically Gaussian, and we show how the asymptotic covariance is specified by the graph Laplacian. For additive parameter noise, we show how the scaling of the asymptotic MSE is controlled by the spectral gap of the Laplacian.