Deniz Ondav scored an own goal and Dan Byrne added another with a header before the break as third-placed Newcastle took control of the match..
But sixth-placed Brighton, who is seeking to play in Europe for the first time next season, narrowed the gap through Ondaf.
Tension prevailed at St. James Park, after Newcastle wasted many opportunities, and Brighton threatened its opponent with a draw, but Callum Wilson and Bruno Guimaraes settled matters in the last minutes, amid the overwhelming joy of the fans..
Newcastle need one of their last two wins to secure a place in the Champions League 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 has 65 points.
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