Haaland was named in a strong starting lineup for Pep Guardiola's side, who required at least a draw to secure top spot in Group G having already qualified for the last 16 along with Leipzig.
The Citizens were not at their best in the first half, though, and they headed into half time two goals down courtesy of a brace inside the first 33 minutes from Lois Openda.
Leipzig had their backs against the wall after the interval as Man City dominated proceedings, and the hosts' persistence in the final third eventually pair off when Haaland placed a side-footed finish beyond Janis Blaswich on the 54th-minute mark.
Phil Foden starred for the Citizens and restored parity with a composed finish in the 70th minute, before Julian Alvarez came off the substitutes' bench to complete the comeback, netting an 87th-minute winner from close range to help Guardiola's men secure first place in Group G with a game to spare.
Haaland, last season's Golden Boot winner in the Champions League, has already scored five goals in as many group-stage games this term, with his latest strike setting two new records in the competition.
Indeed, the Norwegian has now increased his goal tally in the Champions League to 40 after just 35 games, becoming the fastest-ever player to reach that landmark.
Haaland has comfortably eclipsed the previous record held by Ruud van Nistelrooy, who netted 40 Champions League goals in 45 appearances across spells with PSV Eindhoven and Manchester United.
At the age of 23 and 130 days, Haaland has also become the youngest player to score 40 Champions League goals, surpassing the previous record held by Paris Saint-Germain's Kylian Mbappe (23 years and 317 days).
Haaland's latest achievements come just three days after he became the quickest player in Premier League history to score 50 goals when he found the net in the 1-1 draw with Liverpool.
Man City's No.9 has now scored 19 goals in 20 games across all competitions this season, while he has also increased his overall tally for the Citizens to 71 goals in 73 appearances since joining the club in the summer of 2022.
Haaland will look to get his name on the scoresheet once again when Man City play host to Tottenham Hotspur in the Premier League on Sunday.