But both sets of fans were frustrated as they watched their favourites waste a string of chances and ruin a lot of promising attacking play when the ball reached either penalty area.
The first half provided few bright spots, though Hartlepool thought they had got off to a flying start after 12 minutes when Mickey Barron chipped forward for Eifion Williams, but his looping header was well saved by Michele Kuipers.
That was the only real chance the home team had in the first half, and Brighton only managed one more.
First, there was a long-range shot from Richard Carpenter that rebounded off keeper Jim Provett's chest before being cleared. But the keeper made amends when he saved well from an angled shot by Gary Hart.
Provett was in action early in the second half, too, when he did well to save from Robbie Pethick as Brighton started to take control, Leon Knight going close with a long-range shot that whistled inches wide.
But as the chances began to flow, Brighton threw them all away with poor finishing and it wasn't until the last minute that they got another chance on target, Provett again foiling Pethick as the midfield man burst through on the right.
Between these two efforts, Brighton keeper Kuipers did well to block a close-range header from Darrell Clarke and even better to push Marco Gabbiadini's header over the bar after Mark Tinkler's shot was blocked.
"After we lost in the cup down the road at Middlesbrough in midweek, I'm happy with a point from this game," said Brighton boss Steve Coppell. "We played well, and a draw away to a team that scores as many goals as Hartlepool isn't bad going."
Hartlepool boss Neale Cooper said: "A draw and a clean sheet aren't bad against a side like Brighton. But I felt we could have done a little better in this game.
"They are a good side, but we were looking for a win and with more quality in and around the penalty area, I think we would have got it."