Without lone striker Morata, who was replaced by a midfielder in Willian, Chelsea had no outlet up front and City easily picked off possession before attacking in waves.
Alvaro Morata had added to his impressive start with Chelsea by scoring a second-half equalizer after Atletico had taken the lead before halftime with a penalty converted by Antoine Griezmann.
The Brazilian-born Spaniard acknowledged that he grew "a little nervous" as time passed without an agreement between Chelsea and Atletico Madrid.
Chelsea, Manchester United, Manchester City, Arsenal and Everton advance to the last 16 despite fielding virtual second-string lineups.
It was the most explosive moment of a largely sedate lunchtime fixture with Arsenal transformed from the side that collapsed so meekly to lose 4-0 at Liverpool on top of a defeat to Stoke.
Asked later if Chelsea had a new hero called Costa, Chelsea manager Antonio Conte replied in a flash: "Zappacosta."
Hazard is working his way back to full fitness after sitting out the summer with a broken ankle.
Jamie Vardy smashed one back for the hosts from the penalty spot, but Antonio Conte's team held on for the three points.
Not being included in Chelsea's European squad means Costa will not be cup-tied for the tournament if he secures a mid-season transfer.
Everton offered little to show that it could win at Chelsea for the first time in the league since 1994, a run which now spans 23 matches.
Villa fills the void among the strikers left by Diego Costa, who hasn't played for Chelsea this season and isn't in form.
From the Premier League, it will be Chelsea, Manchester City, Manchester United, Tottenham Hotspurs and Liverpool in the 32-team draw in Monaco
With a 2-1 victory on Sunday secured through Marcos Alonso's double, the Premier League champions are finally up and running after a tough couple of weeks.
While Chelsea reignited their Premier League title defense after an opening day loss, Tottenham were condemned to an eighth loss in 11 matches at Wembley since the redeveloped stadium opened in 2007.
Chelsea striker Michy Batshuayi inadvertently headed the ball into his own net in the 82nd minute just after coming off the bench to equalise for Spurs.
The English soccer season started last weekend but Costa is currently in his native Brazil, after claiming he was told in a text message sent by Conte during the offseason that he was not wanted at Chelsea anymore
Chelsea, which won 17 of its 19 home games last season, became the second straight team to begin its league title defence with a loss, after Leicester last season.
The Gunners lift their first silverware of the season as they defeat League champions again to lift the Community Shield.
Referee Singh awarded a penalty to Inter after striker Stevan Jovetic was brought down by Chelsea full-back Cesar Azpilicueta. Some netizens took to Twitter to criticise Singh's decisions and commented on his ethnicity.
The Portuguese having coached Matic earlier in Chelsea admitted that the Manchester club will help him grow better.
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