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.
"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."
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.
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.
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."
