But City were pegged back within six minutes after Mohamed Salah came off the bench and equalised, before Mane converted a spot-kick at the death.
"It's incredible what we have done", Guardiola said.
The English champions had 16 players playing in Russian Federation with the likes of Kevin De Bruyne, Vincent Kompany, John Stones, Kyle Walker, Fabian Delph, Raheem Sterling not due back until next weekend.
City's rivals have strengthened their squads in the close season, but Guardiola is satisfied with his group of players after signing forward Riyad Mahrez from Leicester City for a club-record fee reported to be 60 million pounds ($79 million).
All eyes were on Salah at a wet MetLife Stadium, where the reigning Premier Golden Boot victor and Player of the Year started on the bench following a World Cup affected by a shoulder injury sustained in the Champions League final loss to Real Madrid. He knows the club and the mentality at Bayern and I will be happy if they have success.
Pep Guardiola's side had started the game the stronger despite still missing 15 of the 16 players involved in the World Cup, and they took every opportunity to test Liverpool keeper Lorus Karius, who still looked short of confidence after his costly Champions League errors.
The Spaniard was allowed time off to be with his family at that troubling time and he has now opened up about the experience and about the difficulty in focusing on football.
"The English teams will be favourites for the Champions League".
Asked if City would be the last club in his career, Silva said: "Right now I'm only concentrating on this year and trying to improve on last season".
"A lot of the players went really far in the World Cup". But we are going to adapt.
This game will be filled with some of the best young talents in Europe and it could give one of these fringe players the chance to stake a claim for a place in the starting XI this coming season. His stats for goal involvement over the last three seasons put him among the league's most consistently effective offensive players.