But City played with a lack of urgency throughout much of the half and Bournemouth will be rueing the fact they didn't capitalise on this.
Dani Rodrigues in particular looked dangerous and in the 12th minute he had a shot saved after showing excellent control to tee up the ball and shoot from eight yards.
After 20 minutes great play from City saw Green give a perfect pass to Allsopp, who should have done better from 20 yards despite a good block from Neil Moss.
In the 37th minute Green saw a header tipped over, but Bournemouth hit back with only a brilliant tackle from Richard Hinds preventing Brian Stock from shooting from just yards out.
There was drama in the second half when the Tigers were awarded a penalty after Moss made a hash of a clearance. Young bundled over Allsopp and was immediately sent off. But this time, Green saw his spot-kick saved.
After 65 minutes James Hayter missed a great chance to equalise with a 25-yard shot which went well over.
Stuart Elliott saw a shot palmed on to the bar and Nick Barmby also missed a great chance, but City did just about enough to mark their card in League One.