Stapleton chased down a target of 180 with seven wickets in hand and four balls to spare against Bristol to reach the quarter-finals of the Bristol & District Cup.
They will now be home to Keynsham or Bishopston, who meet on Wednesday.
Holders Knowle went out, losing by seven wickets against Thornbury in a re-run of last season's final. Thornbury now have a home date against Old Bristolians Westbury or Bedminster, who also clash on Wednesday.
Bristol YMCA earned a trip to Claverham - home victors over Timsbury - thanks to a 30-run triumph at Bristol West Indians & Phoenix, skipper Tom Milsom hammering a half century off only 30 deliveries.
Frenchay were ten-run home winners over neighbours Downend, opener Chris Bury (67) and skipper Mike Donaldson (48) sharing a second wicket partnership of 110. Their reward is a home date against Clevedon who bowled out Bristol Pakistanis for 48 en route to a nine-wicket success, Matt Carpenter claiming 5-19.
Quarter-final draw: Claverham v Bristol YMCA, Stapleton v Keynsham or Bishopston, Frenchay v Clevedon, Thornbury v Old Bristolians Westbury or Bedminster. Ties to be played July 2.