Emile Heskey's header early in the second half looked to have given the visitors all three points, but late efforts from Didier Drogba and John Terry turned the match on its head going into injury time.
Villa were not to be denied, though, as Clark headed home at the far post to make it 3-3.
"I am disappointed obviously because I think we used a lot of energy in the second half to come back in the game and try to win," Ancelotti said. "We had a fantastic reaction and played very well with good energy and spirit.
"When we thought the game was won we lost two points in the last situation so, for this reason, I am disappointed because we deserved to win and needed to win. It is our fault because we were unable to maintain concentration.
"In the first half there were a lot of long balls and fouls. The second half was really good, we came back to play our football with intensity, creating a lot of situations to score. We scored two goals. It was good second half."
Ancelotti gave Jeffrey Bruma, 19, his first league start at centre back and, while he praised the youngster's performance, he acknowledged that Chelsea are struggling without their recognised defenders.
"Defensively we lost something," he said. "Obviously I just want to say that Bruma played very well but we are without Alex and Ivanovic - we lost some energy on the high ball.
"We conceded the second goal for this reason. The third was different because we needed to have more attention in the box."
Asked about the club's plans for the January market, he said: "In the centre back I think we have to look at the condition of Alex. This week he comes back from Brazil and we have to look. I don't like to speak here, but we need some players."
Chelsea now sit six points behind leaders Manchester United having played an extra game, but Ancelotti is refusing to admit defeat.
"First they have to win [their game in hand]," he said. "Second, I think the gap is not a light gap in this moment but everything is open again. Obviously we have to beat them here and then you can say something about the title.
"It is not over because we are improving and we can save something this season. We lost confidence because the hard moment was too long, but I think we are improving."
Drogba and Terry appeared to have an argument at the final whistle, but Ancelotti dismissed suggestions of a rift. "Everything is okay," he said.
Gerard Houllier is hoping the results signals the start of a new era for Villa.
"The fact that we managed to get the equaliser meant that we were not letting our heads go down, we were not showing the impression that some teams can have," he told Sky. "I think the equaliser, to me, epitomises the start of the season for us."
Houllier also hit out at rumours of dressing-room unrest at Villa Park following Tuesday's 4-0 thrashing at Manchester City.
"I've read so many things that really stun me, that are nothing to do with the truth," he said.
"Sometimes a player who is not playing calls his agent and his agent calls the press. You can't make a game like that and get a result and particularly react like that if you don't have some togetherness.
"Probably we gave in too early at Manchester City, but that's the past."
He added: "I felt that after the Manchester City game there was a collective awareness that we had to do something together.
"In training, the commitment was different. In the preparation of the game, you could feel and sense things and there was a feeling of togetherness that was good for this game because we knew we were at the foot of a huge mountain.
"In the circumstances that we are in, we knew it would be difficult, but the players were fantastic in terms of effort, in terms of quality of play.
"There was a feeling the fighting was there. When you are in difficulty, the first thing you need is to fight for each other."