Deniz Ondav scored an own goal and Dan Burn added another with a header before the break as third-placed Newcastle took control of the game.
But sixth-placed Brighton, who are looking to play in Europe for the first time next season, have closed the gap thanks to Ondaf.
Tension reigned at St. James Park, after Newcastle squandered numerous chances, and Brighton threatened their opponents with a draw, but Callum Wilson and Bruno Guimaraes sorted things out in the final minutes amid overwhelming cheer supporters.
Newcastle need one of their last two wins to secure a Champions League place for the first time in two decades.
Newcastle have 69 points from 36 games, ahead of fourth-placed Manchester United with 66 points from 35 games. Liverpool have 65 points.
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