It looked like the Yellows would cruise to victory in the second qualifying reply as they led 2-0 at half time thanks to a brace from Jamie McDonald.
But it went all the way as Ashton fought back and equalised in the fifth minute of injury time.
In the end, ‘keeper Tony McMillan was the hero, saving a spot kick in normal time and in the shootout, while all four Yellows penalty takers successful as they won 4-2 on penalties.
It sets up a trip to National League North side Tamworth in the third qualifying round on Saturday week.
Following the 1-1 draw in the original tie on Saturday, in which Warrington missed a hatful of chances, it was the fourth meeting of the two sides already this season.
The first half of this one followed much the same pattern as the weekend, though with fewer chances being carved out.
Half chances for Sean Williams with a shot from range and then Dylan Vassallo’s flick goalwards were the only instances of note in the first 15 minutes and just as Ashton started to get a foothold in the game themselves, the Yellows broke the deadlock.
McDonald collected the ball some 25 yards out from goal, moved forward and fired home a low right foot shot which beat David Carnell, so tough to beat on Saturday, in to the corner of the goal.
It was 2-0 on 26 minutes, McDonald springing the offside trap to latch on to Rory McKeown’s lofted ball over the top, and Carnell hesitated, allowing McDonald to guide the ball round him and walk it in to the empty net.
Ashton lost star striker Matthew Chadwick on 35 minutes, substituted despite not showing many signs of injury, and without him and 2-0 down at half time, their hopes looked bleak.
Even in the early stages of the second half, there didn’t seem much evidence to suggest what would unfold later in the evening.
Tony Gray had a low left foot shot comfortably saved, before substitute Scott Bakkor and McDonald should have done better to carve out an opening on a Yellows counter attack.
Bakkor, brought on for the yellow carded Aboubacar Sanogo, didn’t help much with that situation as he picked up a booking of his own barely two minutes in to his appearance.
Ashton’s lifeline came on 61 minutes though, a soft free kick given against McKeown in the right corner punished as Mason Fawns floated delivery was headed home by Jay Dyche.
That got the home crowd in to the game and Ashton built momentum. A speculator from Fawns almost squirmed through McMillan and in to the goal and then Nicky Platt hit one well from 20 yards what whizzed just over the crossbar.
The Yellows tried to desperately to curb the momentum and settle down the game and just as it seemed that they had done that, Ashton won a penalty.
McKeown won the ball in his own half and in doing so diverted it back to towards McMillan, who rushed out of his goal but was beaten to it by Nick Evangelinos , who went down and the referee had no option but to give a penalty, with McMillan yellow card.
Up stepped Dyche, but McMillan made up for conceding the penalty by pulling off a solid save to his left.
After that, it looked like Warrington could see the game out and they appeared to have done enough until in the fifth minute of injury time, a sloppy pass by Williams enabled Fawns to cross the ball in and Dyche powered a header in to the top left corner for 2-2.
With a tough trip to Mickleover ahead on Saturday, the extra period isn’t what Yellows boss Paul Carden would have wanted, and neither side looked particularly interested in a drab first half of extra time.
But Warrington sensed the tie was there for the taking and they were the only side that looked like scoring in the second period.
Bakkor went close from a Vassallo free-kick, before hammering over the bar when well placed, while Vassallo himself also flashed one across goal after finding room in the box.
A low Williams free-kick was just about held by Carnell and then as time was running out, a flick on by Vassallo just evaded Brad Barnes and penalties it was.
Ashton went first and Fawns blazed over in what would set the tone for the shoot-out.
Yellows captain James McCarten stepped up and gave his side a lead, and when McMillan saved again from Platt, enabling James Barrigan to make it 2-0 after two penalties each, it was all but done.
Williams and McDonald successfully converted their penalties to keep Warrington on the Wembley trail.
Ashton United: Carnell, Smalley, Khanye, Lees, Chalmers, Platt, Roberts (Fawns), Tomsett, Chadwick (Pilkington), Dyche (Woodford), Evangelinos. Unused subs: Eckersley, Ollerenshaw.
Warrington Town: McMillan, Carden (Goulding), McKeown, Barnes, McCarten, Gumbs, Vassallo, Williams, McDonald, Gray (Barrigan), Bakkor (Sanogo). Unused subs: Ascroft, Jennings.