Defending champions City dominated the match throughout but, at 1-1, were still looking for a vital second with the clock ticking down in the second half.
But Alvarez curled home a beautiful strike on 84 minutes to put us 2-1 ahead and Doku sealed the three points as he finished expertly in the second minute of added time.
Earlier, Phil Foden had opened the scoring with a fine right foot finish on 25 minutes before Lois Openda levelled matters on 48 minutes against the run of play.
The result ensured a maximum six points from two games so far in Group G.
Watch our highlights in the video player above.