Michael O'Neill's Northern Ireland side went into the match having beaten Hungary, Faroe Islands and Greece in a flying start to their qualifying campaign.
Romania had secured two wins and a draw from their first three fixtures, though, and now top the group following a hard-fought success in Bucharest.
Chances were few and far between in the opening period, but the visitors did have to survive a scare when Alexandru Chipciu's volley rebounded back off the crossbar with Roy Carroll well beaten.
The hosts predictably continued to dominate possession for large parts of the match, however, and finally made that pay when Papp fired them ahead on 74 minutes.
And any thoughts of a comeback quickly faded away when the full-back headed in a second soon after.
Romania made two changes to the side that won 2-0 in Finland last month, with full-back Papp and 34-year-old playmaker Lucian Sanmartean introduced.
Northern Ireland were without captain Steven Davis, Shane Ferguson and Jamie Ward, so in came Ryan McGivern, Chris Brunt and Niall McGinn, with Gareth McAuley the stand-in skipper.
After a tight first 15 minutes, Brunt was inches wide with a shot across goal on the counter-attack before Chipciu hit the woodwork with a volley at the other end.
Sanmartean showed great skill to dribble past three defenders into the box, but his final pass was a fraction behind Bogdan Stancu, who could not quite get a shot in.
The visitors generally looked fairly comfortable soaking up the pressure, though, and could even have had a penalty themselves when Razvan Rat challenged Conor McLaughlin without getting the ball.
Stancu was replaced by Claudiu Keseru at the start of the second period and he produced a superb pass to set up Rat, but Carroll did well to close the angle and keep out his shot.
Sanmartean caused more problems with another jinking run that ended with him going down in the box, but referee Jonas Eriksson waved play on.
Romania tried to up the tempo as Northern Ireland began to sit deeper and deeper and the breakthrough finally came when Rat fired in a cross that fell to Papp, who thundered his shot high into the top of the net.
Sanmartean then floated in a superb cross for Papp to head home at the far post just five minutes later as Romania clinched all three points.
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