The three-time English Amateur champion was playing in the final stages of the event for the first time, while "The Whirlwind" was bidding for a third consecutive World Seniors title.
Lilley earned his competitive debut at the Crucible after coming through the initial Q-School qualifying in Reading earlier this year.
He built up a 3-0 lead before White, who last year recovered from being 4-0 down in the final, levelled it at 3-3 before Lilley pulled away again for victory.
"I'm a bit speechless, I didn't think I'd get this emotional," he said.
"Jimmy is my hero - he's everyone's hero. I'm just so happy."
Victory means the 45-year-old from Washington, Tyne and Wear, has earned a place in November’s Champion of Champions tournament in Bolton.
Lilley beat Philip Williams, Ken Doherty and Patrick Wallace on his way to the final.