David Silva and Nolito were also on target against the crestfallen former European champions who were spared an even heavier defeat as the visitors hit the woodwork three times.
Borussia Moenchengladbach also took a big step towards qualifying for the group stage with a 3-1 victory at Swiss side Young Boys.
Former European champions Ajax Amsterdam were held 1-1 at home by Russians Rostov while Croatian champions Dinamo Zagreb also had to settle for a 1-1 draw at home to Austrians Salzburg.
Copenhagen beat APOEL Nicosia 1-0 in the evening's other match.
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City's visit to Romania got off to an extraordinary start as Aguero fluffed his lines.
The Argentine, who has missed four of his last five penalties in European competition, saw his first spot kick saved by Florin Nita after eight minutes and then shaved the crossbar with another attempt 12 minutes later.
Even so City found little resistance and took the lead in between when Silva fired home after neat footwork by Raheem Sterling.
Aguero finally found the target four minutes before halftime when he sidefooted the ball in from 18 metres following another chance created by Sterling.
Nolito and Aleksandar Kolarov also hit the woodwork in the first half for rampant City.
OFFSIDE TRAP
New signing Nolito increased the lead for Pep Guardiola's side four minutes after the restart when he beat the offside trap to collect Kevin de Bruyne's pass and rounded Nita for his first goal since his move from Celta Vigo.
Aguero exchanged passes with Nolito before scoring the fourth with an angled drive and he then raced through the defence to complete his hat-trick with a shot off the post.
"We played a really good game. It could have been eight or nine," said De Bruyne. "In the end Sergio scored three goals. If you do that nobody will complain about the two missed penalties."
Two goals in four minutes in the second half put Gladbach in command of their tie.
Substitute Andre Hahn netted within a minute of going on, as the home defence appealed in vain for offside, beforeRaffael's volley was deflected into his own net by Alain Rochat.
Raffael had given the Bundesliga side an early lead but Miralem Sulejmani equalised from close range for the Swiss in the 56th minute.
Christian Noboa floated in a free kick to put Rostov in front, only for four-times European champions Ajax to reply with a Davy Klaassen penalty before halftime.
Andrija Pavlovic's well-taken goal two minutes before halftime, as he latched on to Thomas Delaney's pass and rounded goalkeeper Boy Waterman, gave Copenhagen a narrow lead over APOEL.
Salzburg, who have failed in their last eight attempts to get through the qualifiers, may feel their luck is going to change after the draw in Croatia.
Valentino Lazaro headed Salzburg ahead after a superb run by Wanderson just before the hour although Dinamo, Croatian champions for the last 11 seasons, levelled when Marko Rog won and converted a penalty.
(Writing by Brian Homewood, additonal reporting by Zoran Milosavljevic, editing by Tony Jimenez)
