The two sides could not be separated after 210 minutes of football, with the lottery of penalties ultimately required to decide the tie after two 1-1 draws in the space of four days.
This time, Slough – still unable to name a single recognised striker in their starting lineup – needed a late goal from substitute Manny Williams to force extra time, but it was to be in vain.
16-year-old Purdue – a late call up to the Hendon first team with Tom Lovelock ill with flu – kept out a header from Warren Harris in the ninth minute.
Hendon almost went ahead when Josh Jackman lost possession, giving Zak Joseph the chance to run at goal, but his shot hit the outside of a post.
Simon Dunn saw a 25-yarder well held by Purdue before Matt Lench fired wide from 25 yards after a nice turn and strong run at the Hendon defence.
Despite having a makeshift attacking trio of Lench, Harris and Nathan Webb, Slough had been doing well to pass the ball around Hendon and create opportunities, but the Rebels fell behind on 32 minutes with Daniel Uchechi slotting past Jack Turner and into the bottom corner from around eight yards.
Five minutes later, Dunn hit a post from outside the box but Hendon could well have put the tie beyond Slough before half time, with Turner keeping out a strike from Niko Muir and Dunn clearing off the line a minute before half time.
Hendon had an amazing let-off early in the second half when Purdue slammed a clearance against Lench, but he was able to get back to scramble the ball off the line before saving again from Lench.
The visitors should probably have punished the Rebels again on 52 minutes. After having a good effort blocked a minute earlier, Ashley Nathan-George was thwarted by Turner after being played one-on-one with the keeper.
Slough began to heap the pressure on and Purdue, who looked ill at ease with crosses all evening, fumbled a Lench free kick before grabbing it at the second attempt.
Hendon soaked up the pressure and almost hit Slough on the break, with Turner saving well from Muir down to his left. The striker got in a good position again shortly after, but his finish was well wide.
A double substitution for Slough on 72 minutes saw the introduction of fresh legs in George Wells and the returning Manny Williams, and this gave the home side added impetus in their search for an equaliser.
Mark Nisbet was inches wide with a glancing header from a corner, and as the minutes ticked down it looked like Hendon would escape with a 1-0 victory – but on 89 minutes Williams ran onto a header from Harris, got goal-side of his marker and beat the stubborn Purdue to send the game into extra time.
Purdue saved well from Lench in the first period, crucially gathering the ball after initially spilling it, and Muir twice fired off target in an otherwise uneventful half an hour of football, with both sides clearly tired.
Things started well for Slough in the shootout as Turner saved Sam Murphy’s kick, but Purdue kept out Lench’s effort and then repeated the trick against James Dobson with Slough’s next effort. Michael Corcoran and Casey Maclaren were successful for Hendon, and after Williams hit the bar with his kick, Uchechi stepped up to coolly send Turner the wrong way and win the tie.
Hendon Line up
Daniel Purdue, Jake Eggleton, Sam Murphy, Dave Diedhiou, Rian Bray, Arthur Lee, Daniel Uchechi, Michael Corcoran, Zak Joseph (Keagan Cole 102), Niko Muir, Ashley Nathaniel-George (Casey Maclaren 82). Unused substitutes: Harly Wise, Matt Ball, Kezie Ibe.