The victory was Hamburg's first in three matches and lifted them out of the bottom two and into the relegation playoff place.
Greece international Papadopoulos, who joined Hamburg in January on loan from Leverkusen after a hapless spell at RB Leipzig earlier this season, capped an outstanding performance with a 76th minute header at the far post.
Leverkusen were without injured striker Javier Hernandez and midfielder Hakan Calhanoglu, who has begun a four-month FIFA ban after losing an appeal at the Court of Arbitration for Sport on Thursday over a contract dispute with former club Trabzonspor.
They rarely threatened but did hit the woodwork through Stefan Kiessling in the 85th minute.
Hamburg moved up to 16th in the Bundesliga on 16 points, one ahead of Ingolstadt and seven above bottom side Darmstadt 98.
Leverkusen, who take on Atletico Madrid in the Champions League's last 16 later this month, are in ninth place.
Leaders Bayern Munich, on 45, host Schalke 04 on Saturday while second-placed RB Leipzig, three points behind, travel to Borussia Dortmund.
(Reporting by Karolos Grohmann; Editing by Toby Davis)
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