Argentine striker Tevez, who withdrew his transfer request shortly before the match, failed to shine and Everton took control through goals by Tim Cahill and Leighton Baines in the opening 19 minutes at Eastlands.
Everton were reduced to 10 men when Victor Anichebe was dismissed after 60 minutes following two quick bookings before City pulled a goal back when Yaya Toure's shot was deflected in off Phil Jagielka in the 72nd minute.
City, unbeaten in their last seven league games, also finished with 10 men when defender Kolo Toure was sent off in stoppage-time, also for two rapid yellow cards.
City would have moved a point clear of Manchester United at the top with a win to lead the table at Christmas for the first time since 1929.
United stayed ahead with 34 points from 16 matches, followed by Arsenal on 32 from 17, City on 32 from 18 and champions Chelsea on 31 from 17.
"We are disappointed because we had a lot of chances to score a goal," Mancini told Sky Sports. "But now we must start again. The season is long.
"Maybe in the first half we a started the game not very well, but they only had two chances to score and they scored. Last season it was the same, two chances and two goals."
Everton, without a win for seven league games, began their role as party-poopers in the fourth minute when defender Seamus Coleman crossed for the unmarked Cahill to head home and score for the third successive season at Eastlands where Everton won for the fourth time in a row.
LOW SHOT
Baines started and finished the move that doubled Everton's lead when he curled in a superb low shot wide of England goalkeeper Joe Hart.
Everton got a helping hand in the build-up because City were down to 10 men with Pablo Zabaleta off the field injured before returning with a huge bandage wrapped round his head.
Everton went down to 10 men permanently when Anichebe was sent off on 60 minutes after picking up two quick yellow cards, the second for a foul on Zabaleta.
Tevez worked hard on his first appearance since last week's dramas, but had little impact as former City defender Sylvain Distin marked him tightly until a rasping shot in stoppage-time, his best effort of the match.
City came close to snatching a point when Mario Balotelli burst through after 77 minutes but Everton keeper Tim Howard deflected his effort on to a post.
Monday's match, played in freezing temperatures, was only the third of the weekend's games in the Premier League to be completed because of the wintry conditions across Britain.
Everton manager David Moyes was delighted with his team's display.
"I feel terrific, our resilience in the opening 15 or 20 minutes (has given us) three really good points."
"The stats will show they out-passed us, but we hung in there, it was tough, but we dug in and got it."
