George Thomson's 92nd minute stunner sealed a memorable win at Crabble and ensured Town returned into The National League's top seven.
Town had led from the seventh minute after Jack Emmett's cross was turned into his own net by Scott Doe, but saw their lead evaporate as Jamie Allen and Jai Reason struck in the 79th and 81st minute.
Beck's 84th minute finish put Town back on level terms three minutes later, before Thomson produced an incredible effort in the 92nd minute, cutting in from the left wing and firing past Mitch Walker to seal a memorable win.
After an end to end start to the contest, Town took the lead as Jack Emmett was released down the right from where he tried to pick out Beck at the near post, only for Dover defender Doe to get to the ball first and turn past his own 'keeper.
The flow of the contest was broken up by a series of stoppages after Town had gone ahead, though Dover did have the ball in the net midway through the half, only for National League Player of the Month Kevin Lokko's header to be ruled out for offside.
The hosts threatened again on the half hour mark as Reason got in behind, though James Belshaw was alert enough to close down the danger and ensure the Dover attacker wasn't given a clear sight of goal.
Town finished the half strongly, with Beck's header from Ryan Fallowfield's cross looping onto the roof of the net a reflection of an encouraging first 45 minutes.
While Town opened the half with some strikes off target through Josh Falkingham and Emmett, Weaver looked to the bench on the hour mark, replacing Senior with Howe as Dover pushed forward for an equaliser.
Belshaw prevented any equaliser shortly after when Inih Effiong was baring down on goal under close attention from Fallowfield, but the Dover forward couldn't beat Belshaw for a fourth time this season.
Town's stopper made an even more impressive save five minutes later after Lokko's header had dropped to substitute Alfie Pavey a few yards out, only for Belshaw to make himself big and block the finish.
Belshaw then relied on Fallowfield to keep the hosts out as the full back made an excellent sliding block to prevent Effiong's finish hitting the target.
With 11 minutes to go, Dover found the equaliser their pressure merited as a throw into the Town box dropped kindly to Allen, who turned sharply before firing low past Belshaw.
The tie was then turned on it's head as another long throw caused Town problems, though was eventually cleared, but only into the path of Reason who curled an unstoppable effort into the top left corner.
Despite surrendering the lead they had held since the seventh minute, Town got themselves right back in it three minutes later as Dylan Mottley-Henry released Aaron Williams, who centred for Beck to squeeze the ball past Walker.
Just as it looked as though the teams would share the points, Thomson cut in from the left wing and unleashed a rocket past Walker to seal a memorable late win for his side.
Town: Belshaw, Fallowfield, Senior (Howe 60), Falkingham, Burrell, Thomson, Emmett, Beck, Kerry (Mottley-Henry 75), Muldoon (Williams 82), Kitching
Subs not used: Cracknell, Woods
Goals: Doe Own Goal 7, Beck 84, Thomson 92
Booked: Falkingham, Kitching
Dover: Walker, Lokko, Brundle, Allen, Effiong, McNamara (Modeste 74), Doe, Lewis, Debayo (Pavey 66), Taylor, Reason
Subs not used: Sho-Silva, Adebowale, Worgan
Goals: Allen 79, Reason 81
Booked: McNamara, Reason
Referee: A Dale