"It was important that we resolved this situation with him," City manager Mancini told BBC Sport.
"We are happy for this, but it is the only good news this evening."
Argentine international Tevez, who retained the captaincy for the match, was largely anonymous during City's home defeat by the Toffees.
His U-turn so soon before the match was no doubt intended to have given a boost to his manager and his team-mates.
But the home side failed to break down a resolute 10-man Everton side and missed the opportunity to leapfrog rivals Manchester United and move to the top of the league table at Christmas for the first time since 1929.
Mancini slams Man City defending
"It is very important for everyone; the club, the squad, for him," added Mancini. "The situation has been cleared up and everything has gone in a good way.
"It was important that we stopped talking about this situation. We are happy about that.
"Carlos is a world-class player whose contribution since he joined the club has been invaluable.
"I am pleased that we are now able to focus on pure football matters and are able to look forward to Carlos continuing to play a significant role in the club's progress."
Tim Cahill gave the visitors the lead with a simple header when unmarked in the box, before a superb second from Leighton Baines doubled the Toffees' advantage after 20 minutes.
Yaya Toure pulled a goal back for the home side when his shot was deflected in off Phil Jagielka but Everton, who were reduced to 10 men when Victor Anichebe was dismissed on the hour, held on to record their fourth successive away win over City.
Mancini's side play Newcastle, Aston Villa and Blackpool in the seven days between Boxing Day and New Year's Day and Mancini said: "We must try to win those games.
Can Manchester City bounce back? "We had a big chance to go top after 81 years. It was a fantastic thing. But now we must restart.
"The season is long but we must understand that we cannot sleep when we play."
The Italian believed they had been unlucky not to score after the interval, when captain Tevez, Mario Balotelli and Yaya Toure all came close.
"In the second-half, we played well and tried to win," he said. "We had a lot of chances to score but this is football.
"We were very unlucky, their goalkeeper saved 10 times and maybe there was a penalty but if we don't score we can't do anything."