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Dover led twice and were seconds away from a sensational FA Cup win before eventually bowing out to League 2 Cambridge at Crabble on Thursday night.
Ricky Miller broke the deadlock 20 minutes from time and they looked set to book a home tie with former winners Coventry in round two but Mark Roberts scrambled a stoppage-time equaliser after a Mitch Walker blunder.
Aswad Thomas smashed in a volley to put Whites ahead in extra-time but three goals in the final 10 minutes against tiring Dover defence sent the full-time side through.
The game got underway in a boisterous atmosphere but the only goalmouth action in the opening 10 minutes saw Cambridge midfielder Luke Berry curl a 20-yard free-kick a yard over the bar.
Miller's first chance came after nice interplay down the left from Ross Lafayette and Ricky Modeste with the latter's low cross square across the area fired wide of Will Norris' right post by Miller.
Walker raced off his line to block a low shot from Medy Elito after Berry's fine through-ball and Uche Ikpeazu's glancing header from a high cross was clawed away by Walker soon after.
United spurned a golden chance midway through the half when a low cross from the right reached the far post where Berry slid to fire in a shot from six yards which came off the post and ran across the face of goal before being hacked clear.
Walker was again called upon to keep the deadlock intact when he saved with his feet after the dangerous Piero Mingoia skipped past Tyrone Sterling down the right before crossing low for Elito.
Miller had already sent a couple of long-range efforts off target and seen another blocked when he latched onto a superb pass down the left channel and raced through on goal but saw his first-time toe-poke blocked by the legs of Norris with half-time approaching.
Two minutes before the break Lafayette should have put the hosts ahead when he ghosted into the box unmarked to get on the end of a sumptuous inswinging cross from Modeste, only to head wide from eight yards with the goal gaping.
Jamie Grimes nodded wide from a Sam Magri free-kick four minutes into the second half while Mingoia tested Walker from a narrow angle soon after and Richard Orlu had to head out from under his own bar from a looping Berry header from a set-piece just before the hour.
The deadlock was eventually broken 20 minutes from time when Miller latched onto a Chris Kinnear Jr ball down the right and muscled his way past the last man, with a hint of handball in the process, and into the box before firing low past Norris to send the River End wild and notch his 22nd goal of the season.
Miller was left in a heap by a thunderous challenge from former Gills man Leon Legge soon after but returned to the action gingerly.
Brad Halliday dragged narrowly wide of Walker's post with 10 minutes remaining and seconds later United were within inches of levelling when Elito curled against the underside of the bar but the ball came down in play and Whites somehow scrambled clear.
Another chance went begging when Roberts headed home from six yards after an Ikpeazu cross, only for the linesman to controversially flag him offside.
Miller was withdrawn with an eye on Saturday's league game against Guiseley less than 48 hours away while United laid siege to the Whites goal with Ikpeazu seeing two shots bravely charged down and Legge heading wide from a Mingoia corner.
Roberts was a handful in the box and struck the woodwork yet again in the final minute but in the first of four added minutes Cambridge won a free-kick 20 yards out which Berry took and Walker couldn't hold, the ball rolling to the sliding Roberts to fire into the net and silence the home crowd.
In a flat atmosphere United looked the keener to win it at the start of extra time and Legge glanced a header inches past the post from a Mingoia cross and a minute later Berry nodded another Mingoia cross off target.
Ikpeazu saw a low drive held by Walker before Dover won a corner at the other end in the 101st minute.
Sub Mitchell Pinnock's centre was only cleared as far as Thomas, lurking 25 yards from goal and he struck a thunderous volley into the far corner to spark wild celebrations.
Ikpeazu dragged another effort wide before Walker got down to smother a Berry effort in the second-half of extra-time.
Lafayette curled wide from 20 yards as Dover sought a clincher but with 10 minutes remaining and Dover dropping deeper and deeper, Roberts crossed from the right for Legge to turn home from six yards with the hosts looking dead on their feet.
Pinnock did see a 20-yard free-kick saved by Norris while Berry launched a similar effort over the bar at the other end but Dover hearts were broken with less than three minutes of extra-time remaining when Elito popped up at the far post to turn in a low cross with the Dover defence overwhelmed.
In the dying seconds sub Ben Williamson, once of Gills, strode forward and made an angle to fire under Walker and book a televised home date with Coventry on December 4.
Dover: Walker, Thomas, Sterling, Orlu, Grimes, Magri, Stevenson, Kinnear Jr (Fazakerley 90mins), Modeste, Miller (Pinnock 85mins), Lafayette. Subs not used: Arnold, Jackson. Emmanuel, Howlett-Mundle, Marsh.
Cambridge: Norris, Taylor (Adams 102mins), Dunne, Roberts, Legge, Mingoia, Berry, Dunk (Williamson 66mins), Elito (Coulson 120mins), Ikpeazu, Halliday. Subs not used: Gregory, Long, Maris, Clark.
Attendance: 1,158.