The introduction of Ball at half-time sparked a previously dour game into life. The young target man created space for David McGoldrick to test former England goalkeeper David James within three minutes of coming on.
James was equal to McGoldrick's shot, but could do nothing about Richard Wood's header from the resultant corner, which cannoned back off the crossbar.
Lee Barnard had a chance to give Bournemouth the lead on the hour-mark, but was denied by a smart diving save from Joe Murphy.
The visitors were made to pay for that miss two minutes later when McGoldrick put City in front.
The on-loan Nottingham Forest forward collected a chipped return pass from Ball before advancing on goal and stabbing past former James.