The tie finished 2-2 after 90 minutes but Brian Howard, Liam Walker, Ashley Harris and Luke Rodgers all converted penalties for Pompey, while Lee Barnard saw his effort saved by Mikkel Andersen and Marc Pugh blazed over to send the Cherries crashing out.
Bournemouth took just five minutes to take charge as Shaun MacDonald nodded home Simon Francis’ cross from the right.
However Pompey drew level just before first-half stoppage time, Mustapha Dumbuya’s persistence down the right paid off and Luke Rodgers slid in to convert his cross.
The visitors edged ahead again courtesy of MacDonald’s second of the match, his side-footed effort from the edge of the box whistling straight into his top corner.
Wes Fodgen was inches from extending Cherries’ lead after 66 minutes when he turned on to Harry Arter’s shot and fired it against the foot of a post. And the home side hit back immediately as skipper Howard blasted them level from 25 yards.