Zlatan Ibrahimovic sent Milan on its way in the 33rd minute and Boateng sealed the victory — which had never looked in doubt — with 20 minutes to go.
BATE's best chance came after a glaring error from Mark van Bommel but Milan goalkeeper Christian Abbiati was on hand to spare his teammate's blushes.
Massimiliano Allegri's side should have made its win more comfortable as it wasted several chances, as well as seeing Alberto Aquilani send a powerful effort crashing off the woodwork.
Milan sits top of Group H, level on seven points with Barcelona, which beat Viktoria Plzen 2-0. BATE and Plzen have one point.
Watch video of Boateng's goal below
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