The second-half sending-off of Rodri completed a miserable evening for big-spending City in Bodø, a fishing town with a population of around 55,000 and located north of the Arctic Circle — more than 1000km north of Oslo.
Kasper Høgh scored close-range goals in the 22nd and 24th minutes to set the competition debutants on their way to a first win in the competition.
Jens Petter Hauge sent the home fans wild inside the 8000-capacity stadium by curling into the top corner in the 58th to make the score a scarcely believable 3-0.
"It was an incredible opportunity for us and the feeling is that everything (that can be) going wrong, is going (wrong) in many, many details," City boss Pep Guardiola said, just three days after his side lost 2-0 at fierce rivals Manchester United in the Premier League.
"Bodø were really good and we congratulate them. We cannot say anything else ... I didn't underestimate them."
City replied through Rayan Cherki in the 60th minute but hopes of a comeback were damaged by Rodri - the 2024 Ballon d'Or winner - getting sent off in the 62nd for collecting a second yellow card in quick succession.
They sit seventh in the standings on 13 points, eight behind leaders Arsenal who remain perfect thanks to a 3-1 win at Inter Milan.
Gabriel Jesus struck a first-half brace to fire the unstoppable Gunners into the last 16.
The Brazilian opened the scoring at the San Siro after just 10 minutes before Petar Sucic blasted the hosts level eight minutes later.
Jesus doubled his tally in the 31st and substitute Viktor Gyokeres struck six minutes from time as Arsenal recorded their seventh triumph in as many matches in Europe, meaning Mikel Arteta's side are guaranteed to finish in the top two.
Elsewhere on Tuesday night, Real Madrid (15 points) went second thanks to a thumping 6-1 win over Monaco, with Kylian Mbappe netting a first-half double.
Tottenham jumped to fourth (14) after Cristian Romero and Dominic Solanke goals earned Spurs a 2-0 win over Borussia Dortmund, easing the pressure on manager Thomas Frank.
Defending champions Paris Saint-Germain (13) dropped to fifth after they suffered a 2-1 defeat in Portugal to sixth-placed Sporting Lisbon.
There were also wins for Olympiacos, Ajax Amsterdam and Club Brugge, while FC Copenhagen held Napoli to a 1-1 draw.
With PA.
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