Lewis Elliot, 19, predicted both teams would score in both halves of the Crawley v Carlisle, Hibernian v Aberdeen and Chelsea v Tottenham games.
The bet came in at odds of 6,858/1 when Eden Hazard scored for Chelsea in their FA Cup semi-final.
"I am absolutely delighted," said Mr Elliot. "I went crazy when Hazard scored."
The first two legs of his bet came in as Aberdeen reached the Scottish Cup final with a 3-2 win over Hibs.
Later, Crawley and Carlisle shared the spoils in a 3-3 draw in League Two.
Mr Elliot's win was confirmed at Wembley on Saturday evening, with Chelsea's Willian and Spurs' Harry Kane scoring in the first half of the two London sides' cup clash.
Dele Alli scored for Tottenham just after the break to leave Lewis requiring just one more Chelsea goal to land his bumper bet.
That goal came courtesy of Eden Hazard, who scored in the 75th minute to give the Blues the lead and send them on their way to the FA Cup final.
An apprentice joiner from Hawick has won more than £20,000 from a £3 football bet.