Schieber, starting only his second game for the club, slid home Jakub Blaszczykowski's cross as the English champions again looked out of their depth at European level.
Big-spending City, eliminated before kick-off and out in the group stage for the second season running, finished bottom of Group D with only three points and missed out on the consolation of qualifying for the much-maligned Europa League.
City are the first Premier League team not to win a game in the Champions League group phase and their points tally was the lowest recorded by an English side, eclipsing Blackburn Rovers' record of four points from six games in the 1995/96 season.
Bundesliga champions Dortmund rested Polish trio Robert Lewandowski, Blaszczykowski and Lukasz Piszczek, while Mario Gotze was ruled out along with Neven Subotic, Sebastian Kehl and Sven Bender.
Mario Balotelli, booed incessantly after coming on as a second-half substitute, summed up City's evening when he was booked in stoppage-time for petulantly throwing the ball to the ground when a decision went against him.
"Was it the same old story? Yes. I don't know why," Manchester City goalkeeper Joe Hart told Sky Sports. "I'd love to answer it. We need to address it though.
"The intensity was there but once they scored it disappeared which is a shame as we have showed we are a side that can come from behind. Of course we have let ourselves down, because we haven't even got in the Europa League.
"There was a hunger to win tonight and if that meant Europa League than great. But we have got bigger issues than that. We were up against a weakened team and we had to win."
Schieber's only previous goal for Dortmund had come against lower league opposition in the German Cup.
Dortmund, who had already secured top spot, completed their impressive campaign with four wins and two draws and must be considered contenders to win the title for the second time.
Like City, the Germans were eliminated at the group stage last season but unlike their opponents have clearly learned from that experience.
"I'm glad the coach gave me the chance, that I repaid his trust in me and that it worked out," said 23-year-old Schieber after scoring his first Champions League goal. "We played well and won against a good team."
Dortmund's philosophy is a complete contrast to City with CEO Hans-Joachim Watzke telling Reuters that the club would not exchange its freedom and independence to have an outside investor.
"That is our culture and the culture of our 75,000 members. Being free and independent is German heritage," he said.
The English champions started with Carlos Tevez partnering Edin Dzeko in attack and Sergio Aguero on the bench.
Dortmund had the best chance of an uninspiring first half when Marco Reus cut inside his marker and scraped the outside of the post with a low left-foot shot.
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