The Gunners needed a win to move back into fourth place with one Premier League game to play against Everton on Monday (AEST), but Newcastle did Spurs a massive favour at St James' Park.
White, making his return from a hamstring injury, turned a Joelinton cross into his own net in the second half and Guimaraes added a second late on as the Magpies dominated Mikel Arteta's side.
Spurs will head to Norwich City on the final day knowing a point will almost certainly put them in the Champions League next season due to their goal difference and Chelsea is guaranteed a top-four finish, while wounded Arsenal looks to be destined for the Europa League.
The Gunners made a shaky start, with Aaron Ramsdale fortunate not to gift Miguel Almiron an early goal when he took far too long on the ball following a back pass.
Dan Burn blocked a goal-bound shot from Bukayo Saka, but the Magpies continued to dictate and Ramsdale make an excellent save low to his right to deny Allan Saint-Maximin after the French winger cut in from the left at full tilt.
Arsenal lost Takehiro Tomiyasu to injury before the break and shaken Newcastle defender Fabian Schar was forced off early in the second half, having landed heavily after taking a blow to the head when he was caught by Eddie Nketiah.
It came as no surprise when Eddie Howe's side took the lead after 56 minutes, when White prevented Callum Wilson from getting on the end of Joelinton's cross but turned into his own net.
Wilson volleyed just over the crossbar and had a shot deflected wide by White, before the lively striker came close to an incredible goal with a long-range strike that went narrowly wide.
Guimaraes stuck the knife into Arsenal when he was on hand to tuck home the loose ball with five minutes to go after Ramsdale rushed out to thwart Wilson and only a sharp reaction save from the Arsenal goalkeeper prevented Ryan Fraser from adding a third.