The hosts dominated the first half, in which they spurned some glorious chances, and carved out the best opportunities in the second half but Leeds clung on for a point.
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Brighton were twice denied by the woodwork as they managed 20 goal attempts to the 10 of Leeds but were left frustrated as they drew for the sixth time in their 13 league games.
Leeds, still missing several key attacking players, earned a first away clean sheet of the season and might have claimed a smash-and-grab raid with a few late chances.
Brighton moved above Manchester United into eighth place with 18 points while Leeds remain in 17th spot, three points above the relegation zone, although they could drop into it if Burnley beat Tottenham Hotspur on Sunday.
There were a few boos from around the home stadium from disgruntled Brighton fans at the full-time whistle, a reaction that left manager Graham Potter perplexed.
"It's frustrating but that happens in football," Potter said. "We created chances today and if we maintain that level of performance we'll score goals.
"The fans are entitled to their opinions but I disagree with them. You have to understand the game, who we played and what we did. We are sitting eighth in the Premier League but maybe I need a history lesson on this club."
Potter's Brighton side could have been out of sight by halftime had they put away the chances that their attractive football provided with Neal Maupay the biggest culprit.
He sliced his first opportunity over the bar from right back Tariq Lamptey's low cross and then skewed another effort wide from another Lamptey centre before halftime.
Meslier made another great save late on when he deflected Solly March's low strike against the post.