The visitors nearly took the lead in the 15th minute when Jake Forster-Caskey lifted a cross into the box from a corner and David Rodriguez headed at goal, only to see his effort smack off the crossbar.
Wednesday tried to press and Jeremy Helan had their best attempt after 20 minutes when he collected a long ball and drove at the Brighton defence, but he fired his shot over the bar.
After a bright start from both teams, neither was able to carve out any clear-cut opportunities before the break as quality was lacking in the final third at Hillsborough.
The Wednesday keeper was in fine form to deny Jesse Lingard as he fired an effort from distance, but Kirkland was able to hold the ball to keep his side level.
Wednesday got their winner late in the game on counter-attack. Best drove into the box and fired a low shot which beat the outstretched hand of Kuszczak and nestled in the corner.