Palacio's 10th goal of the season, which came when he met a cross with the right with a delicate touch that left Milan keeper Christian Abbiati standing, lifted Inter to fifth place on 31 points.
Milan have only won one league game in two months and are languishing in 13th place on 19 points, only five away from the relegation zone, after a poor performance in which Sulley Muntari was sent off in stoppage-time following a late scuffle.
The defeat could have been worse had Inter been given a penalty five minutes before the break when Palacio was brought down by Cristian Zapata, but struggling Milan still ended 2013 in the most miserable fashion.
(Reporting by Terry Daley, editing by Ed Osmond)
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