 Solution

False. Take $X=[0,1]$, $Y=S^1$ and the surjection $f(t)=e^{2\pi it}$. The interval is compact and connected, and the circle is Hausdorff. But $f^{-1}(1)=\{0,1\}$ is a disconnected two-point subspace. Thus a continuous surjection need not have connected fibers.