Joe Willock and Callum Wilson scored the crucial goals which lifted the Magpies into third place in the league standings and kept alive their hopes of qualifying for the Champions League next season.
Both Newcastle and Manchester United have 50 points from 27 games, but Newcastle have the better goal difference in the race for a top-four finish.
Speaking after the game, Willock said, "I feel like we have to believe. We've got a brilliant team, brilliant manager, and brilliant environment for us players to work in, so I feel like there's nothing stopping us."
Newcastle dominated play at St James' Park, narrowly missing the target from several excellent chances. Their persistence was rewarded in the 65th minute when Allan Saint-Maximin's headed cross found Willock for a close-range header. Substitute Wilson then put the game to bed in the 88th minute, heading in a freekick cross from Kieran Trippier.
The victory was sweet revenge for Newcastle after losing 2-0 to Manchester United in the League Cup final in February. "We played really well," said Willock. "I feel like we deserved the victory and it was electric today in there, and I feel like we fed off (the fans), and I feel like we owed them one after the cup final. We didn't take that lightly."
Newcastle's performance was outstanding, with the team creating plenty of chances and putting pressure on Manchester United's defense from kickoff.
Manchester United goalkeeper David de Gea was forced into a double save in the 16th minute, diving to punch away Alexander Isak's header and then stopping Willock's close-range shot with his body. Willock sent another fabulous scoring chance soaring just over the bar in the 40th minute.
Speaking after the game, Newcastle manager Eddie Howe said, "I think we've had some moments like this since the turn of the year, where we haven't scored the goals that maybe we should have done. First half was another example of that really where we were frustrated not to lead. But we needed patience, we felt if we performed the same way in the second half, the goal would come. Thankfully it did."
Manchester United were missing the influence of midfielder Casemiro, who was serving a suspension, and defender Aaron Wan-Bissaka due to illness.
However, the return of Marcus Rashford was good news, at least, after the striker sat out England's Euro 2024 qualifying games with a knock. Despite their absences, Manchester United were expected to put up a strong fight, but Newcastle proved to be the better team on the day.