Eubanks lost the first set to the Greek fifth seed and then trailed two sets to one but he powered back to win 3-6, 7-6(4), 3-6, 6-4, 6-4.
It is another crushing disappointment at the All England Club for Tsitsipas, who has never been beyond the fourth round.
Tsitspas started strongly, breaking his 43rd-ranked opponent twice in the first set but he was edged out in the second-set tie-break.
Two more breaks in the third set put the Australian Open finalist back in control but Eubanks broke in the ninth game of the fourth set to level.
The American unleashed a searing single-handed backhand down to the line to break Tsitsipas in the first game of the decider.
Tsitsipas returned the favour in the sixth game to level things up but Eubanks produced another moment of magic on the backhand to break again, cupping his ears to the crowd.
Eubanks held his nerve when he served for the match, saving two break points before producing a devastating forehand to seal the deal.
He soaked up the atmosphere of an enthusiastic crowd, making a heart gesture towards them with his hands.
"It's like I'm living the dream right now," he said. "This is absolutely insane."
The American described grass as the "stupidest" surface to play on last month.
But he has since won the Mallorca tournament on the surface before hitting form at Wimbledon.
"Those words will never come out of my mouth for the rest of my career," he said in reference to his comments.
"The grass and I, we've had a very strenuous relationship over the years but right now I think it's my best friend."
Before arriving at the All England Club, the 27-year-old American's record at the Slams had been modest.
Of his eight main draw appearances he had lost in the first round six times. He had tried and failed four times to qualify for Wimbledon.
