Leeds were the first to threaten, with striker Ross McCormack having an early shot saved in the fifth minute.
Despite a cagey opening period, Bournemouth grew into the game and came close when Lewis Grabban picked up the ball 30 yards out, dropped his shoulder and unleashed a wild effort that flew just over the bar.
The striker's kick was saved by substitute Darryl Flahavan, who dived to his left to keep the ball out and ensure the score remained 0-0 going into half time.
Leeds made their extra man count straight after the break when first-half villain McCormack got his right boot onto a low cross to divert the ball past Flahavan in the Bournemouth goal and put his side into a 1-0 lead.
Having withstood heavy pressure, 10-man Bournemouth launched a surprise comeback after Grabban followed a flick-on to head home and level the scores.
The away side's spirit was finally overcome by Leeds in the 80th minute as Poleon picked his spot to volley home and that's the way it stayed as Brian McDermott's men secured three hard-fought points with a 2-1 victory.