Ajax's defeat by Real Madrid meant City would have finished third in Group D had they beaten the German champions.
But Dortmund posed the greater threat, forcing a series of saves from Joe Hart before Julian Schieber slid in to score from Jakub Blaszczykowski's cross.
City crash out of Europe without a win in any of their six group games.
Their haul of three points is a meagre return given the vast resources at Roberto Mancini's disposal, and is the lowest by an English club in the 20-year history of the competition.
On this evidence, a vast improvement is required if they are to make any impact. If it had not been for some superb saves by Hart, City would have lost by an even greater margin to a depleted Dortmund side without many of their stars.
Jurgen Klopp's side were already through as group winners so the onus was on the visitors to chase victory, especially with Ajax behind early on in Madrid.
But Mancini's men created little in the first 45 minutes, and their only effort on goal was a bouncing shot by Edin Dzeko that Roman Weidenfeller pushed round his post.
Carlos Tevez worked hard and covered a lot of ground but, without the injured David Silva, City were lacking a spark in attack.
Dortmund also began slowly before creating a host of chances either side of half-time.
Hart, City's hero when they snatched a point against Dortmund at the Etihad Stadium in October, came to their rescue before the break when he got his fingertips to a Marco Reus drive and dived to keep out Ivan Perisic's curling shot.
The home side went closer still when Mats Hummels was just unable to turn in an inviting free-kick from Blaszczykowski, who came on for Marco Reus at the interval.
Edin Dzeko headed straight at Weidenfeller as City tried to respond but the chances kept coming for Dortmund, with Hart tipping a swerving Perisic volley over the bar and reacting well to beat away Kevin Grosskreutz's fierce strike.
