Some two minutes elapsed before Eddie Howe's side threatened the upset, with Yann Kermorgant drilling a shot straight at Boaz Myhill.
West Brom brushed off that early scare and went close themselves through Liam O'Neil, who saw his back-post header drift narrowly wide of the mark.
Both teams had good chances to move ahead prior to the break, as Myhill thwarted Tokelo Rantie before West Brom record signing Brown Ideye failed to beat Lee Camp from a good position.
Bournemouth broke the deadlock on 49 minutes through O'Kane, who latched on to Junior Stanislas's pass to slot past Myhill early in the half.
The clash looked to be petering out as the visitors struggled to fashion the opportunities necessary to get back on level terms.
It was Georgios Samaras who helped restore parity, however, as the second-half substitute saw his shot deflect off the leg of Elphick five minutes from time, but Wilson struck moments later to send Bournemouth into the fifth round.