Hanover, trailing 1-0 to an Elias Kachunga goal, won a free kick in the Paderborn half deep into stoppage time and goalkeeper Ron-Robert Zieler went up to join the attack as they threw everything forward.
Paderborn cleared the ball which fell to Stoppelkamp on the edge of his own area. Facing his own goal, he chested it down, turned and thumped it down the pitch into the unguarded goal, the ball landing just inside the Hanover half and bouncing six times before it trickled into the net.
Schalke 04 drew 2-2 at home to Eintracht Frankfurt despite playing the last 20 minutes with nine men after Kevin-Prince Boateng and goalscorer Julian Draxler were sent off, while the visitors had Slobodan Medojevic dismissed in the 85th minute.
Troubled Hamburg SV, playing their first match under new coach Josef Zinnbauer after he replaced Mirko Slomka, held champions Bayern Munich to a goalless draw at home in a bad-tempered game while Augsburg beat Werder Bremen 4-2 and Hoffenheim won 2-0 at VfB Stuttgart.
Promoted Paderborn, Hoffenheim and Bayern all have eight points from four games.
Alex Meier and Marco Russ put Frankfurt 2-0 ahead in less than half an hour at Schalke, before Eric Maxim Choupo-Moting pulled one back with a Panenka-style chipped penalty after Medojevic handled in the 40th minute.
Draxler, celebrating his 21st birthday, drew Schalke level five minutes into the second half before the cards began to flow.
Ghana midfielder Boateng was dismissed for a second bookable offence just after the hour, followed nine minutes later by Draxler after a clash with Carlos Zambrano.
Medojevic was dismissed in the 85th minute when his trip on Tranquillo Barnetta earned him a second yellow card.
Bayern Munich coach Pep Guardiola had made four changes to the team which beat Manchester City in the Champions League, but was then forced to make another when Arjen Robben suffered a muscular problem in the warm-up, according to German media.
Bayern goalkeeper Manuel Neuer gave supporters some anxious moments with his forays upfield and nearly came a cropper late in the game when he slipped around 40 metres from his goal, lost possession and then handled.
He was booked but angry Hamburg players demanded that he be sent off for denying them a clear scoring chance.
Thomas Mueller had an opportunity to give Bayern a late winner when he pounced on a mistake by Johan Djourou but his shot was deflected over.
Anthony Modeste and Tarik Elyounoussi gave Hoffenheim a win at Stuttgart, who are bottom with only one point. Tobias Werner and Tim Matavz scored in the last 15 minutes to wrap up Augsburg's 4-2 win over Bremen
(Writing by Brian Homewood in Berne; editing by Toby Davis)
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