William Osula opened the scoring inside 12 minutes after a catastrophic error from Brighton goalkeeper Bart Verbruggen, who was caught far off his line and could not recover as Osula intercepted and finished into an empty net. Dan Burn headed home a Guimaraes corner to double the lead before half-time.
Brighton responded after the break through Jack Hinshelwood, who combined with Danny Welbeck before firing past Nick Pope, and Yankuba Minteh — the former Newcastle United winger — was presented with a clear chance to level but sent his effort over the bar. Harvey Barnes, introduced from the bench, settled the contest late with a composed finish.
Guimaraes, making his 150th Premier League appearance, was named player of the match and spoke with conviction about the dressing room atmosphere before kick-off.
"It felt very good. In the changing room before the game, we knew today was important for us, chasing European football for next season. The points are crucial. We keep dreaming and believing."
He was candid about the second half, in which Brighton had the better of play.
"It was a tough game. Brighton are a good team. Today, for everything we have done, we deserve the win. They had a lot of chances. In the first half we were better, but in the second half they were better. I am glad we could keep three points."
On what Newcastle need over their final three matches, Guimaraes was direct.
"We are looking up, not down. We can still play European football next season. We must win the next three games."
The Brazilian midfielder was returning from a ten-week absence with a hamstring injury — the first significant fitness setback of his career. He started for the second consecutive week and completed the majority of the 90 minutes before being replaced late on.
"I feel amazing, very good after ten weeks out. It was the first big injury of my career. I am still chasing my best physical level, but today I did a good performance."
Newcastle move up to 13th, four points from the Conference League places. They travel to Nottingham Forest next weekend in what could prove the pivotal fixture of their season.