City were 2-1 down when spot-kick specialist Caballero read Falcao's intentions to make a save that energised the Etihad Stadium.
Aguero then equalised and although Falcao’s wonderful chip on the hour mark put Monaco ahead again, City rallied and scored three late goals to win a thrilling encounter.
Pep Guardiola’s side now hold an advantage in their Champions League last-16 tie with the Ligue 1 leaders – and Aguero pinpointed his compatriot's save as the turning point.
“We came back and stood up after Willy saved the penalty," he said. "We firstly dropped our heads when the ref didn’t give us a penalty, and Monaco took advantage of the situation and scored.
“After Willy saved the penalty in the second half we got revitalized, people were behind us supported us and we could give everything we have.
“It's not over but it's important that we're two goals up, if it was one obviously it would be more complicated."
Aguero scored twice in the second half – the first of which was City’s 200th goal in European competition.
He also saw a penalty appeal waved away in the first half when he went down under the challenge of Danijel Subasic.
Aguero has started City’s last two matches having spent the previous three games on the bench, and although he didn't score at Huddersfield, his phenomenal strike rate was in evidence on Tuesday with a quick-fire second-half double.
"It's always important to score goals," he said. "I like it when they're decisive goals and today they were."