Kieran Lee threatened for the visitors in the opening stages, but saw his powerful effort go just wide of the post.
Bournemouth were back on level terms just past the half-hour mark through Marc Pugh, who headed in from Lewis Grabban's cross.
It did not take the Owls long to regain their lead, as within two minutes Sam Hutchinson glanced in a header from close range following a long throw-in.
Andrew Surman came close to scoring for the hosts in the final minute of the first half, but saw his close-range effort saved by Kirkland.
The Cherries almost took the lead just moments later, but Grabban turned Simon Francis's low cross wide of the post.
In the 63rd minute Wednesday were reduced to 10 men after Hutchinson picked up his second yellow card of the afternoon.
The hosts took control of the game and Grabban forced Kirkland to push his effort behind, before Kermorgant's free kick came crashing off the bar.
In the sixth minute of stoppage time Wednesday made sure of the three points when Chris Maguire slotted home from Best's pass following a quick break.