The new Premier League season came to the Etihad, and they marked such an occasion with fireworks and flares while also standing for a minute's applause in the 22nd minute to commemorate the victims of the terror attacks in Barcelona and Manchester.
And by the time the smoke had cleared from the pyrotechnics, the hosts were in control. Otamendi got the first sight on goal, but his tame long-distance effort was well collected by Jordan Pickford.
At the other end, Dominic Calvert-Lewin began his lively combative evening by racing onto a ball and flashing an early shot across the face of goal.
However, from then on, chances came few and far apart for Koeman's men. Pickford made a few smart saves to stop his team going behind, with Manchester City attacking in numbers to pile on the early pressure.
Jagielka made a couple of early blocks to deny Sergio Aguero, and the Toffees' best chance came when Tom Davies hit Calvert-Lewin's pass first-time and the ball whistled past the post.
Pickford pulled off another great save as Otamendi took aim from outside the box, before Aguero tried to chip the ball over the keeper, Jagielka making the crucial clearance despite the ball not being goalbound.
The experienced defender once again stood in Aguero's way inside the box, poking the ball out from under the striker's feet. The momentum was swinging City's way and the hosts had their best chance when Silva smahed the ball off the base of the post.
Only moments later, with the crowd at the Etihad still reeling from Silva having hit the post, Wayne Rooney stunned the hosts. Calvert-Lewin slid the ball into the path of the former United man in the middle of the box to strike. Ederson could not stop the ball sliding underneath him and going in off the post for Rooney's 200th Premier League goal.
But it went from bad to worse for Manchester City, as Kyle Walker was dismissed on his home debut for the club. The right-back first went flying into a sliding tackle on Leighton Baines and was brandished a yellow card. Only two minutes later, he was shown a second booking for backing into Calvert-Lewin to further complicate his team's chances.
Guardiola was still complaining about the red card to assistant referee Michael Oliver as both teams came out for the second half. The hosts dominated possession, evidently more eager to go forward in search of a goal.
Everton were happy to sit back and wait for a chance to break, showing little weakness as they put on a display of solid defending. Sigurdsson made his Everton debut, forced to defend with Manchester City piling on the chances.
Bernardo Silva scuffed a volley wide before Sterling fired a couple of efforts over the bar, the pressure cranking up and the home fans showing their frutration at the Etihad. Pickford then got down low to deny Danilo's left-footed shot brilliantly as we entered the final ten minutes.
But when all hope seemed lost, City found their equaliser. Danilo pinged the ball in from the right and Holgate, who had done almost everything right up to that point, headed the ball straight at Sterling. The winger hit a thumping first-time volley into the bottom corner.
Only minutes after the score was levelled, the number of players on each team were also. Another debatable red card as Schneiderlin went charging into Aguero from behind but mostly getting the ball.
Manchester City pushed forward in the final seconds, but as the final whistle was blown it was a point apiece. Manchester United are now the only team out of last year's top six to win both opening fixtures.