Talbot opened the scoring in the 16th minute, snapping up a rebound after Bournemouth goalkeeper Shwan Jalal had been unable to hold onto a Jamie Lowrie free-kick.
Chesterfield had plenty of chances to add to their lead but Jack Lester missed two clear openings before Talbot struck again, Donal McDermott opening up the Bournemouth defence for the striker to score from 15 yards.
After the break Brett Pitman and Jason Pearce had headers cleared off the line for Bournemouth and Steve Fletcher struck the bar with a header.
They pulled a goal back in the 75th minute when a centre from Liam Feeney caused chaos in the Chesterfield defence and Danny Hollands scored from 15 yards.
And in the 89th minute Bournemouth almost levelled when Alan Connell had a shot blocked by Robert Page.