Julian Schieber scored the only goal of the game for Borussia Dortmund to leave Roberto Mancini's side bottom of Group D with just three points from their six matches.
Having already been eliminated from the Champions League knockout stage, City had a chance to take third place and qualify for the Europa League Round of 32.
But despite Ajax losing 4-1 to Real Madrid, City failed to take advantage and will now have to turn their attention to domestic football - starting with the Manchester derby this weekend.
"We wanted to qualify for the Europa League but we didn't take our chances which makes it very difficult," admitted Mancini afterwards.
"We were in a tough group but if you make mistakes in early games as we did you will not qualify."
Dortmund had the first chance after 12 minutes but Oliver Kirch's attempt was saved by City keeper Joe Hart.
Edin Dzeko, partnering Carlos Tevez in the Manchester City attack, put in one of the rare energetic performances from a City player first putting in a dangerous cross, which Marcel Schmelzer scrambled to clear.
Dzeko came close again nine minutes later with a shot which Dortmund keeper Roman Weidenfeller managed to push away.
Dortmund did not rest on their laurels and tested Hart again in the 36th minute as the keeper got his hand on Marco Reus's shot and deflected it off the outside of the right post.
Hart was kept the busier of the two goalkeepers, stopping an Ivan Persic long-range effort just before the break, then a volley from the player just before the hour mark, followed by a low strike from Kevin Grosskreutz a minute later.
But the hosts finally grabbed the lead in the 57th minute as Blaszczykowski raced down the right side and crossed into the box where Schieber slid in to poke the ball past Hart.
Tevez managed to slip through three defenders but his shot was parried by Weidenfeller in a rare moment of City inspiration.
Dortmund coach Juergen Klopp was surprised that his side were able to dominate the group as they did.
"We didn't think we would take 14 points though I knew that we would be tough," he said.
"But to win games against those types of teams is amazing,"
Hart could not hide the disappointment felt by the English side, however.
"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," the goalkeeper said afterwards.
"Of course we have let ourselves down, because we haven't even got in the Europa League."
"There are still great teams left in the Champions League. But I think we have earned some respect. You can't just look past us. We definitely are looking forward to what comes next."
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