Both the teams went into the half-time with no goals scored, but City's Edin Dzeko opened the account for his club in the 57th minute of the game. 11 minutes later, Aguero scored his first goal for Manchester City when he swooped in from a close range and made it 2-0.
The stoppage time saw Aguero scoring again after man of the match David Silva had made it 3-0 a few minutes before that.
In an ominous warning to his rival clubs, a visibly ecstatic City manager Roberto Mancini told Sky Sports, "He needs two or three weeks to be 100 percent maybe. For me he is a fantastic striker, he is young. I'm delighted after this performance but I think in the first half Swansea played very well."
Aguero, a £38 million signing from Atletico Madrid has almost had no time to mingle with his new team-mates and came on just after the first goal. He made an immediate impact and almost netted with his first touch to make one of the most memorable debut cameos English football has seen in recent years.
Meanwhile, Swansea came back to back England's top-tier for the first time since 1983.