Charlie Daniels' solitary strike was enough to sink the Lilywhites, who end their first season back in the third tier with a dismal run of one victory in 10 matches.
A game of few chances was thrust into life within the opening five minutes of the interval when Luke Clark fouled Donal McDermott inside the box.
Referee Simon Hooper pointed to the spot and the subsequent penalty was calmly dispatched by Daniels for his second goal of the season.
Wes Thomas ensured a nervy ending for the Cherries after he lacked composure and slipped after being played through one-on-one with North End goalkeeper Andreas Arestidou.
Graham Westley's men conclude the season in 15th place, having won only three times since his arrival as manager in January.