The first opportunity of the match fell to Blackpool's Matt Phillips, who unleashed a left-footed strike from the edge of the penalty area after five minutes, but Tomasz Kuszczak was equal to the effort.
The hosts continued to apply the pressure in the early stages, with defender Alex Baptiste squandering another chance to open the scoring from close range.
Andrew Crofts tried to force the issue for the struggling Seagulls on the quarter hour mark with an ambitious volley, but his 20-yard effort sailed wide of the right-hand post.
Isaiah Osbourne picked up the game's first booking after 24 minutes for a foul on Will Buckley, before Elliott Grandin went close once again for the Tangerines with a mishit header from 12 yards.
Gary Taylor-Fletcher was involved in a heated confrontation with Adam El-Abd in the 34th minute, before attempting an audacious lob from eight yards, which flew harmlessly over the crossbar.
With half-time fast approaching, Baptiste fancied his chances from distance and fired a superb strike from 30 yards, but he saw his effort clear Kuszczak's crossbar.
Brighton's El-Abd earned himself a booking after the break with a foul on Tom Ince, before home defender Ian Evatt received one of his own for his protests.
Blackpool continued to dominate in the early stages of the second half and had a penalty appeal turned down by the referee for an alleged push on Kirk Broadfoot.
Despite the home side's dominance, it was the visitors who broke the deadlock in scrappy fashion after 56 minutes with an Ashley Barnes toe-poke from inside the six-yard box.
Tom Ince went close for the Tangerines shortly after the hour mark as they attempted to cancel out the Brighton goal - which had come completely against the run of play - but he scooped his close-range effort over the crossbar.
Ian Holloway's men grabbed a deserved equaliser in the 72nd minute, when Grandin got his head to a Matt Phillips cross to send the ball past Kuszczak, who could have done a better job of dealing with the danger.
Grandin then forced the Polish goalkeeper to push the ball over his crossbar for a corner with a superb 25-yard effort.
As the game approached its conclusion, Craig Mackail-Smith attempted a spectacular effort, meeting a Buckley cross with an overhead kick from the centre of the area, but Blackpool goalkeeper Matt Gilks was equal to it, ensuring the match ended all square.