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Dover's FA Cup dream was ended in the second round at London Road on Sunday but the margin of defeat did scant justice to their efforts.
Ivan Toney's stunning seventh minute goal gave Posh a half-time lead but it was not until the closing quarter that the League 1 side were able to breathe easier.
Their second goal – from Frankie Kent – came just seconds after Jack Munns had gone close to an equaliser for the National League underdogs.
Mo Eisa completed the scoring and will have been grateful to have done so having earlier missed three clear-cut chances.
Dover suffered a big setback ahead of kick-off when 10-goal top scorer Inih Effiong was ruled out by a knee injury.
Oscar Gobern and Aaron Simpson were also sidelined, while loan striker Ruel Sotiriou was recalled by parent club Leyton Orient after the midweek defeat by Maidenhead.
Boss Andy Hessenthaler made a big call in goal picking 36-year-old birthday boy Lee Worgan ahead of Yusuf Mersin, who had been No.1 for the previous eight matches.
Will De Havilland, Kurtis Cumberbatch, Ricky Modeste and Steven Rigg all earned recalls.
De Havilland resumed his central defensive partnership with Scott Doe in a move which saw Paul Rooney move to right-back.
Posh – 56 places above Whites in the football pyramid –are the country's leading scorers with Toney and Eisa as their spearheads it was easy to see why.
Former Newcastle striker Toney needed just seven minutes to make his mark with a strike that could well be the goal of the round.
Keeper Christy Pym's goal kick was headed up into the air by De Havilland and when it dropped Toney met it full on the volley to fire home from 35 yards.
The same player immediately saw another effort blocked by Rooney, while Worgan did well to race out of his area and halt Eisa with a fine sliding challenge.
The front pairing had Dover at full stretch again on 21 minutes when Eisa broke quickly before crossing to Toney whose first time volley was just wide.
To their credit Whites remained competitive after the goal, but they created few chances.
Jai Reason was just inches away from getting on the end of a Munns free-kick and saw another shot deflected for a corner, but home keeper Pym was largely untroubled.
Ricky Modeste won a free-kick in a good position but with men waiting in the box, Bobby Joe Taylor curled the ball well over.
Posh should have doubled their lead on 39 minutes when Frazer Blake-Tracy squared the ball to Eisa eight yards out but he screwed his shot wide.
That was followed by Dover's best chance of the half two minutes later as Taylor swung over a free kick and Rigg stooped to head wide.
Posh had one last chance before half-time and it was another good one as Marcus Maddison blazed over after being played through by Toney.
Eisa had another excellent opportunity during the opening seconds of the second half but with Toney waiting for a pass, he opted to go it alone and Worgan did well to turn his shot aside.
The pair continued their personal duel on 58 minutes when Toney teed up Eisa but Worgan got down brilliantly to his right to palm the ball away.
Whites gained some encouragement from a couple of corners in quick succession and on the hour De Havilland got up well to meet Cumberbatch's cross but his header was saved diving to his left by Pym.
That chance came during their best spell of the game and there was a growing sense of frustration among the home fans who felt Posh needed that second goal.
Home hearts were certainly in mouths on 78 minutes when Munns' shot from the edge of the box kissed the outside of he post but just seconds later it was all over for Whites as Posh doubled their lead.
Maddison played a corner into the near post and Kent stole in to flick the ball home.
Worse was to follow for the visitors on 85 minutes as Eisa finally got his goal, slotting home from six yards after substitute Serhat Tasdemir had squared the ball for him.
Hessenthaler claimed afterwards that the goal was yards offside.
Tasdemir could have made it 4-0 with the last kick of the match but dragged his shot wide.
Dover: Worgan, Rooney, Taylor, Doe, De Havilland, Woods, Munns, Cumberbatch (Jeffrey 82mins), Reason (L'Ghoul 65mins), Modeste (Ratti 70mins), Rigg; Subs: Mersin, Lokko, Bedford, Hinchiri.
Peterborough: Pym, Butler (Woodyard 67mins), Beevers, Kent, Thompson, Reed, Blake-Tracy (Mason 75mins, Maddison (Tasdemir 82mins) Ward, Toney, Eisa; Subs: Chapman, Bennett, Kanu, Jones.
Attendance: 4,239 (500 Dover fans).
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