The game had been carried over from Tuesday due to a thunderstorm and Aguero made up for lost time with his second treble of the campaign after his hat-trick against Steaua Bucharest in the play-off round.
The Argentina striker, currently serving a domestic ban, struck twice in the first half and once in the second before giving way to Kelechi Iheanacho, who netted City's fourth.
City manager Pep Guardiola said the delay had no adverse effect on his players, quipping: "We were at home, sleeping with our wives.
"This was the most important game in the season to now," he added. "Considering the opponent, the pressure that we had, we played really, really good."
With his former club Barcelona, as well as Celtic, lying in wait in Group C, Guardiola had warned his players that a strong start in the competition was imperative.
City reached the semi-finals for the first time last season, while Guardiola, who has won all seven of his games at the helm to date, is bidding to win the tournament for a third time.
It was a third successive defeat against City for Andre Schubert's Gladbach, who lost twice to the English side in last season's group stage.
"We had to get everything right, be very courageous and attack high up with high intensity," said Schubert. "We didn't manage to do that. We couldn't stop City's build-up and we didn't show quality in defence. That's why Man City won so clearly."
After Tuesday's postponement, Gladbach left tongue-in-cheek letters of apology to the bosses of the fans who stayed on in Manchester for an extra day on the seats in the away section.
The only change to the two teams announced for Tuesday's game was City's inclusion of Ilkay Gundogan in place of David Silva, who was rested "as a precautionary measure".
Gundogan debut, having been sidelined by a knee injury since signing from Borussia Dortmund for 20 million pounds, it was Gundogan's first outing in a City shirt.
City attacked the game in a manner akin to their slick first-half display in Saturday's 2-1 win at Manchester United and it took them less than nine minutes to go in front.
Aleksandar Kolarov fizzed a cross into the box from the left and Aguero darted in front of Andreas Christensen to stab home, ending a five-game Champions League scoring drought.
Visiting goalkeeper Yann Sommer saved sharply from Gundogan after an incisive move involving Fernandinho, Kevin De Bruyne and Aguero, while Fernandinho shot narrowly wide.
Gundogan won the penalty that brought City's second goal when he was caught on the back of the leg by Christoph Kramer, although contact may have occurred just outside the box.
Aguero sent Sommer the wrong way, taking his tally for the campaign to eight goals in all competitions. Sommer also had to beat away a shot from Gundogan, but Gladbach improved after Schubert sacrificed Kramer for Julian Korb and switched from a back three to a back four.
A spectator up to that point, City goalkeeper Claudio Bravo got down smartly just before half-time to parry a crisp Lars Stindl effort.
City initially had to work a bit harder for their chances in the second half. But the hosts raised their level with 15 minutes remaining and after Sommer had pulled off sharp saves to thwart Raheem Sterling and Aguero, the pair combined for City's third in the 77th minute.
Sterling's pass down the inside-left channel took Korb out of the game and Aguero rounded Sommer before rolling in his hat-trick goal.
Leroy Sane came on for his home debut and made a quick impression by teeing up Iheanacho to smash home in stoppage time.