Kenny Dalglish recalled Lubomir Moravcik to the side in the place of the injured Mark Viduka.
Beye set up Mark Viduka, who completed a confident win for Kevin Keegan's men with a sure-footed second-half strike.
The visitors were also unlucky not to earn a penalty after Mark Viduka appeared to be tripped by Scimeca.
BBC: SPORT | Football | Premiership | West Brom 1-2 Middlesbrough
The Sunday Express takes it a step further, insisting that Mark Viduka, Alan Smith, Harry Kewell and Paul Robinson are all on the verge of quitting.
Newcastle had opportunites to grasp at a lifeline in the first half, with Obafemi Martins firing off target and Mark Viduka having a shot cleared off the line by Carlos Cuellar.
Former Middlesbrough striker Mark Viduka was introduced for his first appearance of the season after 65 minutes, coming on to a predictable chorus of jeers from the fans who once supported him.
He has stood firm and resisted the temptation to raise the money at a stroke by selling off the family silver in the form of Alan Smith, Mark Viduka and Paul Robinson.
BBC: SPORT | Football | My Club | Leeds United | Leeds' final countdown
Shearer threw on Obafemi Martins and Mark Viduka to partner Michael Owen in a three-man attack and it nearly paid dividends but Martins blasted over from close range with three minutes left.
Leeds' Mark Viduka - one of eight British-based players in Frank Farina's starting line-up - held the ball up brilliantly before delivering an overhead pass into the path of the breaking Bresciano who volleyed home.
Shearer expressed concern that Newcastle would struggle on the pitch in the Championship unless the team's off-field problems were swiftly rectified, allowing a new manager to strengthen a squad whose departures have included Michael Owen, Obafemi Martins, Mark Viduka, Habib Beye and Sebastien Bassong.
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