Trailing 1-0, Budd won four frames on the bounce helped by breaks of 53 and 64.
Foster pulled one back but Budd secured the seventh thanks to a tricky pink on the side cushion.
The left-hander said: ‘I’m so delighted to have won this. Hopefully this will be a confidence booster in events to come.’
In the earlier rounds, Budd defeated Gary Polkinghorne 4-2, ex-professional Danny Auld 4-3 and Chris Robinson 4-2.
Budd is searching for a sponsor before having a crack at Q School and the chance of turning professional. The end-of-season event carries a £1,000 entry fee.
He said: ‘I won’t be doing Q School this year but maybe next year depending on how my game is and because it is expensive.
‘If anyone is looking for any advertising, I would be happy to put a company logo on my waistcoat.’