Raheem Sterling opened the scoring for City, but the Citizens later found themselves 2-1 and 3-2 down, but a brace from Aguero and goals from John Stones and Leroy Sane gave Guardiola's side the edge in the tie.
Arguably the key moment in the match occurred in the 50th minute, when Nicolas Otamendi fouled Radamel Falcao - the scorer of two of Monaco's goals - and Monaco were awarded a penalty.
A successfully converted spotkick would have given the Ligue 1 side a 3-1 lead, but a brilliant save from Caballero thwarted the Colombia international.
"It was very important, it kept us in the game to be honest," Toure said of the save.
"His penalty save and another one was unbelievable.
"We have a fantastic team togetherness and with this kind of win we showed it.
"We can go far in this competition but we don't want to start to dream from now. First there is the second leg to play.
"When we fight so hard to the end it is unbelievable."
Toure also singled out Aguero for praise, after the Argentine returned to the City starting line-up in place of the injured Gabriel Jesus.
Aguero twice equalised, with a 58th-minute strike that burst past Danijel Subasic in the Monaco goal and a sweetly-struck volley 13 minutes later.
"That's why I say competition in the team is very important. It makes the team much better," Toure continued.
"I think you can see he played much better – full of desire.
"That's why we want him, that's why the manager wants him. He is there for the team."