The hosts broke away to open the scoring after 21 minutes with Jeff Hughes placing the ball into an empty net after Bournemouth goalkeeper David James lost his bearings.
It was 2-0 to County after 32 minutes when Simon Francis left his pass to James short and Yoann Arquin nipped in to fire home.
A moment of magic by Donal McDermott saw Bournemouth pull a goal back in the 51st minute when his exquisite flick put Josh McQuoid through to finish.
The Cherries levelled on the hour after Marc Pugh was tripped by Neal Bishop inside the area and Harry Arter sent County keeper Bartosz Bialkowski the wrong way from the penalty spot.
Jamal Campbell-Ryce looked to have won it for the hosts when he squirmed a penalty under James after 84 minutes but Tubbs squeezed home another equaliser in the 85th minute to share the points.