An opportunist strike from Zlatan Ibrahimovic and a spectacular effort from midfielder Kevin-Prince Boateng left Milan with seven points from three games after a wildly unpredictable Group H match.
Milan could have had a hatful of goals, yet should have found themselves behind against the side whose full name is Borisov Automobile and Tractor Electronics and who have still to win a Champions League match in nine attempts.
With Milan alternating moments of inspiration and sloppiness, the game turned on a one-minute spell in the first half.
A dreadful defensive slip-up by Milan left Borisov playmaker Renan Bressan with a clear run on goal in the 32nd minute but the Brazilian took too long over his shot and Christian Abbiati was able to turn it wide.
"You have to make the most of those chances, you only get one against teams like Milan," Bressan told reporters.
One minute later, Milan went ahead when Ignazio Abate's dangerous low cross was only partially cleared and Ibrahimovic raced in to rifle home a first-time shot.
The volatile Swede had nearly given Milan a spectacular start when he saw goalkeeper Aleksandr Gutor off his line and tried to catch him out from 40 metres, his effort going just over.
He also wasted another chance when he nonchalantly side-footed over the bar after Antonio Cassano's floated cross found him on the right.
Alberto Aquilani also hit the foot of the post in a lively Milan opening.
The goal seemed to make Milan overconfident as Ibrahimovic tried to flick the ball over a defender's head and missed another chance when he clipped the ball delicately across the face of the goal from Cassano's incisive pass.
Borisov, who finished with three draws and three defeats in their only previous Champions League campaign in 2008/09, were glad to hear the half-time whistle.
The Belarus champions for the last five seasons came out with renewed hope and they briefly controlled the match early in the second half, the ineffective Mateja Kezman wasting their best chance on the break when his shot was blocked.
Milan, who had 24 shots but only five on target, continued to perform in fits and starts.
Ibrahimovic seemed to sum it all up just before the hour when he did everything right, cutting inside and making space for his shot, only to sky a wild effort high into the stands.
Robinho's introduction for Cassano gave Milan more life and he nearly made an immediate impact, cutting inside and firing wide.
Milan finally settled the game when a long cross-field pass found Boateng and the Ghanaian - who somehow escaped a yellow card for kicking an opponent who was on the ground - scored with a rasping 25-yard shot which went in off the underside of the crossbar.
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