The Argentine, who joined in July for a reported City record of $64 million, entered the fray on the hour after Edin Dzeko gave the hosts the lead and was soon on the score-sheet himself when he tapped in Micah Richards' low cross in the 68th minute.
Three minutes later the former Atletico Madrid forward turned provider when he superbly hooked a ball back from the byline for David Silva to slam home.
Aguero then fired in a mesmeric fourth goal from distance in stoppage-time.
"He needs two or three weeks to be 100 percent maybe," City manager Roberto Mancini told Sky Sports in an ominous warning to City's rivals.
"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."
The Welsh side did indeed make a decent fist of the opening exchanges in their first ever Premier League game but the gulf in class soon began to show.
City's victory was the only home win of the opening round of English top-flight matches and their fans, bereft of silverware for 35 years before May's FA Cup triumph, will now be dreaming of snatching the league crown off Manchester United next May.
RICHEST CLUB
The world's richest club took their time to break the deadlock but were solidly on top from the middle of the first half and could have gone ahead when Silva and Gareth Barry rattled the bar while Yaya Toure blasted wide.
Bosnian striker Dzeko eased any Eastlands nerves when he tucked the ball away in the 57th minute after Swansea's busy Dutch keeper Michel Vorm, also making his league debut, parried Adam Johnson's shot following a superb driving run by Silva.
Aguero, having had no real practice in a City shirt after finishing the Copa America last month, came on just after the goal and almost netted with his first touch as he made one of the most memorable debut cameos English football has seen in recent years.
"There is a great bond in this group, I'm feeling the love," City keeper Joe Hart said.
Stephen Dobbie's second-half effort was Swansea's best although it barely troubled Hart. They will take consolation in the fact that they will face much easier tests this term even if everyone at the club knows they are in for a long hard season.
Their style has been compared to Blackpool, who became the neutral's favourite last term for their attacking endeavour but were still relegated straight back to the Championship.
"I said to the players I was very, very proud of them tonight," Swansea manager Brendan Rodgers said.
City have their eyes fixed on the opposite end of the table and Aguero's explosive debut will have assured fans they have an ideal replacement for Carlos Tevez, who was not picked on Monday after asking weeks ago for a move which has not yet transpired.
