Defending champions City, who came into the game on the back of losses to Newcastle in the Carabao Cup and Wolves in the Premier League, looked in complete control in the first half.
Phil Foden gave them a deserved lead with a difficult shot on the bounce following a cutback by Rico Lewis, who impressed in a central midfield role.
Leipzig scored with their only shot on target, as Lois Openda raced onto a through ball before slotting the ball past Ederson via the inside of the post.
But it was all City from then with Foden hitting the post with a free-kick and Erling Haaland missing a host of chances.
Five minutes after coming off the bench, Alvarez lifted a perfectly weighted shot into the top corner from the edge of the box following a pass by Doku.
The Argentine then found Doku in space and he sped away to score a late third and seal the points.
City have now equalled the longest European unbeaten run by an English club – level on 16 with Leeds (1967-1968), Tottenham (1967-1972), Chelsea (2018-2019) and West Ham, whose run started last year and is still going too.
Pep Guardiola’s side will be relieved to get back to winning ways as they prepare to visit title rivals Arsenal on Sunday.
“I will not judge my team on results,” Guardiola told TNT Sports. “Both games were not the best results, but it is not easy when both teams defend with 10 in the box.
“If you think I will doubt my players, you are wrong. I rely on these guys unconditionally. We go to Arsenal, rest and hopefully in May we will arrive in the competitions being there.”
City looked unbeatable in the opening weeks of the season as they started to build on last season’s Champions League, Premier League and FA Cup haul.
The vibe was slightly different coming into this. Had they gone three games without a win, something that never happened in that Treble season, Sunday’s Premier League game against Arsenal would feel even bigger.
But Guardiola’s subs saved the day.
Haaland had a frustrating night against his favourite opponents. The Norway superstar had scored 11 goals in his previous six appearances against Leipzig, including five in City’s 7-0 win in March, but failed to find the net with six efforts at the Red Bull Arena.
But often when Haaland doesn’t deliver, Alvarez does. He only had six touches – but two of them produced goals.
He scored an absolute beauty himself and then returned the favour to Doku who ran through to net his first Champions League goal.
“We played a fantastic game in all departments. Everyone was perfect,” added Guardiola.
“The substitutes all came off the bench with smiles on their faces, not complaining because they did not start.”
This also marked City’s first win in Leipzig. The clubs have now met in three consecutive Champions League seasons, with Leipzig winning and drawing the others.
