The Swede fired the Serie A champions ahead in the 33rd minute, just after Renan Bressan had missed an outstanding chance to give the Belarus side a shock lead.
Kevin-Prince Boateng calmed nervy Milan with a superb long-range strike in the 70th minute after Borisov had looked capable of snatching an equaliser.
Milan have seven points from three games and look certain to qualify from Group H along with Barcelona. Borisov have one point.
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