O’Kane gave Bournemouth the lead eight minutes into the second half, gobbling up the rebound after a Ryan Fraser shot had been pushed out by Pompey goalkeeper Phil Smith.
Wes Thomas should have opened the scoring for the home side mid-way through the first half when he blazed a header over from Simon Francis’s cross.
Smith produced a stunning fingertip save to keep out Norwich loanee Andrew Surman’s 20-yard curler on the stroke of half time.
At the other end Tom Craddock had Bournemouth goalkeeper Ryan Allsop scrambling across his line after 18 minutes with a long range shot which fizzed narrowly wide.
Surman almost doubled the Cherries’ lead in the 65th minute when he fired wide from a Ryan Fraser cross, while Smith was at his best again 14 minutes from time to turn another fierce Surman drive wide of the post.