Snainton were the better side in the first half and opened the scoring on 25 minutes through a cracking long-range shot from Rob Holt.
Early in the second half Sherburn levelled, a Mark Bean free-kick was met by Rich Malthouse, who lifted his shot over the keeper, only for Carl Sample to tap the ball in on the goalline.
Striker Ryan Collings then restored Snainton's lead with a close-range strike, but in the 80th minute Matty Whitehall levelled to take the game into extra-time.
Whitehall put Sherburn ahead early in the opening period of extra-time with a superb header, Collings then had a goal disallowed, but two minutes from time Snainton looked to have set up a penalty shoot-out when an Andy Worgan corned sailed in directly,
Sherburn refused to give up hope and claimed the winner with the final kick of the game when sub Jack Pinder volleyed in from 12 yards after fine work from Adam Spaven
Sherburn boss Andy Adamson said: "It was a very good game and we played superbly as a team, so much so that it is impossible to single out one player for man of the match.
"Snainton started well and were the better team in the first half, we came back well and were stronger in the second half and in extra-time I think our extra fitness paid off as we kept going to the end."
His opposite number Andy Holt added: "It was a great game and our man of the match was Jack Heelas."
The Fishburn Park v Itis Itis Rovers quarter-final was abandoned due to an unplayable pitch with the visitors holding a 6-1 advantage before the ref called a halt to proceedings.
Reserve League leaders Edgehill Reserves strolled to a 9-0 home win against lowly Ayton Reserves.
A Robbie Scarborough hat-trick along with efforts from Ryan Link and Benny Davis fired Edgehill into a 5-0 half-time lead.
A strong display from the hosts saw them add another four after the break through Carl Hepples, Ollie Parker, Joshua Fergus and another by Davis.
Youngsters Scarborough, Parker and Fergus were the star players for Edgehill, with Nicko Dunn man of the match for Ayton.
Scalby Reserves slipped to a 3-2 loss at home to Newlands Reserves.
The visitors opened the scoring through Lee McLoughln, but Taylor Jordan's fine strike levelled the scores after great link-up play with Dave Oxley.
In-form striker Sam Foy made it 2-1 to Scalby after the interval with a neat finish before Brandon Catherall gave away a penalty, which was converted by Aiden McCallion.
Ryan Matson scored the winner after an error from the stand-in Scalby keeper.
Dave Young was named man of the match for Newlands for a commanding midfield display.
Scalby player-boss Josh Bowmaker said: "Credit to my whole team for a great performance with only having bare 11 and a stand-in keeper."