Mancini's men gave a lacklustre performance at the Liberty Stadium and were punished by substitute Luke Moore's 83rd minute winner for the Swans.
With United cruising to a 2-0 win over West Bromwich Albion, City were pushed off pole position by their bitter rivals for the first time since October.
Despite a frustrating afternoon in south Wales, at the end of a week which also featured a 1-0 Europa League loss to Sporting Lisbon, Mancini remains confident City are still in the right frame of mind to take the title fight to the final weeks.
"We shall fight against them until the end. We have another 10 games to finish this Championship," he said.
"The season is long and the situation changes every week. It is important that we stay together. But this result changes nothing.
"It is better to stay at the top, of course, but we are now a point behind. It does not change our target. I don't see it as a critical game. I think this championship will be decided with two games to go."
Mancini admitted Swansea had given his side a tough time, with Joe Hart saving an early penalty from Scott Sinclair, but he was disappointed City were unable to convert any of their own chances.
"It has been a difficult last four days for me. But I don't think we deserved to lose the game," Mancini said.
"Swansea played very well for the first 30 minutes, they played very good football.
"After that, in the second half we had control of the game, but we missed three or four chances and that is the problem we have at the moment. After that we conceded a stupid goal. We cannot concede goals like this.
"We could be tired because after seven months, when you play three games in one week, it is difficult."
Mancini hauled off Gareth Barry after 36 minutes but claimed that his decision was based on a mystery injury rather than his performance.
"Gareth has had a problem for three weeks. They had a lot of possession for 30 minutes and we needed another striker," Mancini said.
"It is not because he played bad. He has had a problem but I don't know where. I know it is not a big injury."
Meanwhile, Swansea boss Brendan Rodgers was left to salute his side's latest impressive showing in their first season back in the top flight.
"Over the course of the game some of our passing and ball retention and movement was phenomenal," he said.
"We got one goal but we also missed a penalty too. We could have had one or two more and I think the players deserve every credit they get for the tactical and technical ability of their game.
"There have been some wonderful results over many years at this great club but certainly it was a terrific performance.
"When you take on a team like Manchester City, with all the resources they have got, and you dominate the game like we did, it fills you with great pride.
"As a manager I am still learning and always will be. I have the L plates on! But I thought it was a defining moment when we played them up there at the start of the season.
"I congratulated the players afterwards although we had just been beaten 4-0. For an hour of that game they showed that we can play our style and technical game and even hurt the biggest opponents.
"We just needed to continue with that and mature as the season went on. I think we have achieved that and this confirms that."
Source: AFP
Source: AFP