Two near-identical goals, one scored by Bernardo Silva and the other assisted by him, put Pep Guardiola's side within touching distance of the quarterfinals.
Bernardo put City in front on 29 minutes, losing his marker at the back post to head home a pinpoint Joao Cancelo cross.
The same move resulted in Bernardo heading back across goal for Gabriel Jesus to tap in on 65 minutes and put City in a commanding position with two away goals.
The tie took place at the Puskas Arena in Budapest, Hungary, rather than in Monchengladbach, due to German authorities putting travel restrictions on the UK amid the spread of a COVID-19 variant.
"The first leg is always more difficult. Up front, we have to be more clinical," Guardiola said. "The one-against-one [for Gabriel Jesus] against the keeper. In this competition, you have to be perfect to make sure that you go through."
Gladbach, who only had one attempt on target in the entire game, had their best chance through Alassane Plea's spectacular backflick-volley just past the hour but were otherwise boxed in their own half.
"Manchester City were really good in possession, which meant we did a lot of running tonight," said Gladbach's Florian Neuhaus. "We had the odd chance in transition here and there but did not execute them cleanly."
"We wanted to be courageous with our limited chances and I felt we did that well at times. We have to be braver in the second leg."
They have now lost three of their last four games in all competitions while City have won their last 12 away games in all competitions, setting a new record for an English top-flight side.
City are 10 points clear of second place in the Premier League, in the Carabao Cup final and in the quarterfinals of the FA Cup as they chase an unprecedented quadruple. They also extended their unbeaten run to 26 games in all competitions and are bookmakers' favourites for the Champions League as well.
"It is a big challenge to recover the players and make sure they are in the best possible condition. If people want to say we are favourites, it's okay -- we will accept it."