The visitors created the first opportunity of the afternoon when David Lopez got past his maker and let fly, but it drifted wide of the upright.
Early chances were otherwise scarce, but the hosts were in front before the half-hour mark through Bakary Sako.
The Frenchman found the net from 12 yards after playing a neat one-two with Kevin Doyle to cap off an impressive team move.
Wolves suffered a setback shortly before half-time when Karl Henry was red-carded for violent conduct, but things would get worse for the hosts before the break.
Brighton hit back just before the interval, as Bruno swung in an inviting cross from the right and Craig Mackail-Smith fired into the top corner to make it 1-1.
Wolves were back in front just on the hour mark through Tongo Doumbia. The Malian international found the bottom corner with a sublime strike from 25 yards.
Once again, Stale Solbakken's side failed to hold on to a lead and were pegged back to 2-2 in the 72nd minute when William Buckley latched on to Liam Bridcutt's pass and fired home to draw the Seagulls level.
Things looked bleak for the hosts in the 88th minute when Dobbie put Brighton in front from the spot, but there was late drama to come.
Sako whipped in a cross two minutes into stoppage time and Johnson leaped high in the box to head home and rescue a point at the death.