Kieran Millar put the home side in front after just two minutes when he converted Ciaran Summers's cross from close range.
Ross Brown spurned a good opening for East Fife towards the end of the first half when he skewed his shot wide of the target.
After the break, Ross McMillan headed over for Stenhousemuir before Allan Walker saw a shot from the edge of the box saved for East Fife.
Jonathan Page levelled the tie up with three minutes remaining when he headed home Walker's delivery to take the game into extra time.
However, their parity was short-lived, and it was Jamie McCormack who gave Stenhousemuir the lead once again four minutes into extra time when he turned the ball home from close range.