Bakary Sako opened the scoring for Stale Solbakken's men when he fired the ball past Tomasz Kuszczak after some superb build-up play.
Craig Mackail-Smith pulled Brighton level on 43 minutes, before Karl Henry was shown a straight red card for a reckless challenge on Liam Bridcutt on the stroke of half-time.
Wolves were back in front against the run of play on 61 minutes with a 25-yard strike from substitute Tongo Doumbia, but a 12-yard drive from Will Buckley levelled things again.
Brighton took the lead for the first time with a Stephen Dobbie penalty in the 89th minute, but Johnson had the final say when he headed the ball in from a Sako free-kick just a minute later.
The Seagulls enjoyed lengthy spells of possession in the opening 15 minutes, but Wolves fought their way back into the match and, after 22 minutes, it was the hosts who opened the scoring. Sako received the ball from Kevin Doyle at the end of a superb passing move and slotted it past Kuszczak from 12 yards.
After withstanding considerable pressure from the home side, Mackail-Smith pulled the Seagulls level in the 43rd minute. The frontman received the ball from Bruno Saltor in the penalty area, before firing into the top-left corner of the net from 12 yards.
The goal prompted Solbakken to punch a hole in the roof of the home dugout, but he probably should have saved his outburst for a couple of minutes later as things were about to get much worse for the Wolves boss - Henry receiving a straight red card for a poor challenge on Liam Bridcutt.
Brighton started the second half brightly as they tried to make their one-man advantage count, and twice went close with efforts from David Lopez and Dobbie.
Ten-man Wolves then regained the lead against the run of play in the 61st minute - substitute Doumbia making the most of an advantage played by referee Scott Mathieson to fire in a superb 25-yard effort.
Brighton pushed to pull themselves level again and, on 72 minutes, they succeeded with a low 12-yard drive from Buckley.
However, there was more drama to follow when Christophe Berra was controversially judged to have handled the ball in the penalty area on 89 minutes, giving Dobbie the opportunity to fire the ball past Ikeme from the penalty spot.
Johnson then rose superbly to meet a Sako free-kick and head the ball into the back of the Brighton net to earn his side a hard-fought point and score his first goal for Wolves.