City showed fantastic resistance to earn a draw after recovering from a disastrous nine-minute spell just before half time.
Wayne Rooney's 200th Premier League goal and Kyle Walker's controversial dismissal for two quick-fire yellow cards left City in a perilous position at the break.
But despite the numerical disadvantage, City produced an inspirational second-half performance to force a leveller through substitute Sterling's brilliantly-struck volley after 82 minutes.
City had much the better of the play early on but could not convert the pressure into chances - Silva's shot against the post was the closest attempt - while the Toffees' first effort on target had Ederson picking the ball out of the net.
At that moment City were facing an uphill task, but the job became mountainous without Walker. The wing-back's first yellow came for a sliding challenge on Leighton Baines, and the second left the Etihad crowd bemused after he was cautioned for a collision with Dominic Calvert-Lewin.
That stinging spell of play left the home fans with a feeling of deja vu at the break. The corresponding fixture last season saw City draw 1-1 after two missed penalties, while the 4-0 Goodison Park defeat was a season nadir.
And although there were new faces on display - Ederson and Walker were making their home debuts and Rooney, Michael Keane and goalkeeper Jordan Pickford played in a revamped Everton side - the frustration endured.
Before Rooney's opener City had chances to take the lead.
Nicolas Otamendi's powerful shot was parried to Aguero, but the forward's cross was headed over the bar by Phil Jagielka.
City were really stepping up the pressure midway through the half, and after fine work from Jesus and De Bruyne, Aguero just failed to get his shot away under pressure from Jagielka.
Aguero then turned provider, curling a sublime ball with the outside of his boot to Silva, but the Spaniard's driven shot thudded back off the post.
The fine margins in these games became all too clear when, after Leroy Sane lost possession on the left flank, the ball broke for Dominic Calvert-Lewin, who crossed for Rooney to finish through Ederson's legs.
Despite Walker's dismissal, City still dominated the play in the second half.
Sterling came on for Jesus, and Silva, De Bruyne and Aguero remained prominent in the play going forward.
The key pass in the final third was hard to find against stubborn Everton resistance and when the Blues did force an opening through crisp interplay between De Bruyne and Aguero, Sterling's shot slid over.
The breakthrough finally came on 82 minutes and in some style. Sterling was perfectly placed to smash home a volley following Mason Holgate's defensive header.
City pushed hard for the winner - Everton themselves were reduced to 10 late on when Morgan Schneiderlin was sent off - with Bernardo looking threatening every time he had the ball at his feet, but the game ended all square.
Matchday app star man – Raheem Sterling
Boss Pep Guardiola made just the one change to the side that defeated Brighton, bringing Sane in for Danilo in the left wing-back role, with Ederson and Kyle Walker making their home debuts. City started with Kompany on the left of the back three, with Stones central and Otamendi on the right, but Kompany and Otamendi switched sides midway through the first half. Sterling came on at the break for Jesus as Guardiola reacted to Walker's dismissal. Danilo (for Stones) and Bernardo Silva (for Sane) were second half substitutions as City tried to force the play.
What it means?
It's early days in the Premier League but City move to four points after two games following the opening day win at Brighton.
City travel to Bournemouth on Saturday for the lunchtime kick-off, before the home clash with Liverpool a week after.