Facing Leipzig in the competition for the third straight year, City dominated from the start, Phil Foden slamming in the opener after some impressive lead-up play from teenage wing-back Rico Lewis.
Clearly outclassed in the first half, Leipzig struck back immediately after half-time, Lois Openda scoring after a sweeping counter attack to equalise.
But with Leipzig focused on talisman Erling Haaland, who scored five in 60 minutes the last time these sides met, it was the Norwegian's understudy who scored the crucial goal, curling in City's second just five minutes after being introduced.
Doku sealed the win with a goal in injury time.
City forward Foden praised Alvarez's "magical" goal, telling TNT sports "he's someone we look to for the big goals.”
"He came on and changed the game. He's a really important player for us.”
City boss Pep Guardiola praised his "perfect" side, saying "we took the game with a lot of passes and when a team is able to do this, very nice things are going to happen to you.”
"We played a fantastic game in all departments.”
City came into the game after two losses in a row for the first time since January.
Guardiola told the post-match press conference the victory put his team's recent struggles "in perspective”.
"People say 'oh, you lost a game, what happened?' but you need to put it in perspective, after what we've won in the last years, we're still here and we have got this passion.”
Leipzig manager Marco Rose called the result "a deserved win for City, even if we would have wished to defend that 1-1 until the end.”
"We'll continue to learn from the best."
Foden and Lewis impress
Bullish in the first half of Saturday's 2-2 draw with German champions Bayern Munich, Leipzig were timid in the opening half, repeatedly losing the ball on the rare occasions City gifted it to them.
Foden and Lewis terrorised Leipzig down the right flank, the two linking for the opener 25 minutes in.
Lewis found an onrushing Bernardo Silva, who played it back for the 18-year-old to find Foden, who hammered the his shot into the grass, the ball looping past 'keeper Janis Blaswich and into the net.
City showed their quality, cutting through the stagnant Leipzig defence, Haaland going close twice while advancing from the left to close the half.
Rose made no changes to his stuttering side at half-time but Leipzig emerged a different team, drawing level just three minutes after the break.
Yussuf Poulsen threaded a pass to a sprinting Openda, who kept City's Manuel Akanji at bay with one arm while squeezing his shot in off the post as a frustrated Guardiola kicked the turf, cupping his face in his hands.
With Leipzig happy to sit and counter and City equally happy to control possession, Foden looked the most likely, and he nearly caught out Blaswich with a free-kick that bounced back off the crossbar.
Foden was replaced with 11 minutes remaining by Alvarez and the Argentinian World Cup winner would prove crucial, collecting a pass from fellow substitute Doku and unleashing an inch-perfect shot into the top corner.
Doku got a goal of his own as the clock wound down, finishing off a City counter to seal the win for the visitors.
The victory was City's first in three attempts in Saxony, having drawn and lost in previous visits to Leipzig in the past two Champions League campaigns.