Hamburg snapped a two-game winless streak to reach sixth place as Rudnevs scored the only goal of the match in the 50th minute.
Stuttgart, meanwhile, came into the game on the back of a 2-0 Europa League defeat to Lazio and a third straight loss in all competitions sees the Reds struggling down in 14th place.
Hamburg were the first side to threaten as Rudnevs headed wide from close range following good work from Rafael van der Vaart.
Stuttgart's best chance of the first half came in the 16th minute as Raphael Holzhauser's free-kick was deflected wildly towards Hamburg keeper Rene Adler, who smothered at his near post.
Serdar Tasci's header was smothered by Adler as play resumed but Stuttgart were no closer to finding the back of the net.
Hamburg then grabbed the lead in the 50th minute as Dennis Diekmeier crossed from the right to find Rudnevs, who blasted a volley home from the middle of the box.
The hosts nearly equalised in the 69th minute, though, with Shinji Okazaki's cross finding Vedad Ibisevic, but Adler was there to deal with his header.
Hamburg pushed for a second decisive goal in the 79th minute but Tomas Rincon's shot from distance sailed wide of the mark.