Keith Andrews had a good chance to open the scoring for Brighton in the early stages, but he could only fire an effort over the crossbar after collecting the ball in space.
The hosts completed a turnaround before the break when Tommy Spurr advanced into the box to find Cairney, and the midfielder added to his assist by striking a low effort home.
Blackburn applied most of the pressure in the second half, but were denied a third goal of the evening Craig Conway's deflected shot struck the post.
Tomasz Kuszczak was then forced into action to deny Conway, who was given space to attempt to target the bottom corner from distance.
It looked like Blackburn had found a late winner in the 86th minute when Rhodes, who had been bundled to the floor by two Brighton defenders, stepped up from 12 yards to beat Kuszczak with a penalty.
However, there proved to be one more twist as substitute Dale Stephens found space inside the box two minutes later to head in Lingard's cross and earn a point for Brighton.