Abraham, the home favourite, took the bout 115-112, 116-111, 112-115.
Murray, 33, fighting for a world title for the fourth time, started well and, after Abraham improved, rocked the 35-year-old in the eighth round.
But Abraham, with impeccable defence and accurate flurries, had too much class for the St Helens fighter.
Even a point docked from Murray in the 11th round for using the shoulder - he had previously received several warnings for holding - had no bearing on the final result.
Abraham, who completed a fifth successful defence, improved his record to 44 wins and four defeats, while Murray suffers a third loss to go with 32 wins and one draw - the four fights he has failed to win were all for world titles.
Though he matched the Armenia-born German in the opening three rounds, getting close any time Abraham looked like unloading, Murray was pushed backwards from round four onwards and took a huge right in the seventh.
Murray, though, was able to absorb Abraham's best and landed a shocking right of his own in the eighth, one that put the champion in danger for only time in the bout.
The Briton was unable to capitalise, though, and Abraham kept him it at bay for the remainder, drawing the point deduction he had been asking of the referee for most of the fight.
And, when the decision came, Murray only had the favour of one of the three American judges.
Though two of his previous world title fights ended in debatable points decisions - a draw with Germany's Felix Sturm in 2011 and a unanimous decision loss to Argentina's Sergio Martinez in 2013 - Murray this time has less right to feel aggrieved.
Meanwhile, on the night Anthony Crolla became WBA lightweight champion, Tyrone Nurse beat Chris Jenkins for vacant British super-lightweight title and Ryan Burnett beat Jason Booth for vacant bantamweight crown.
Britain's Martin Murray fell short of taking the WBO super-middleweight title from Arthur Abraham, losing a split points decision in Hannover.