Rafael Benitez's side also trail Premier League leaders Manchester United by 11 points, and goalkeeper Cech has placed the blame for their predicament squarely at the feet of their forward players.
While Benitez has defended Fernando Torres, who has now gone eight games without a goal, the interim boss opted to go with Demba Ba up front at the Liberty Stadium knowing his side needed at least two goals, and appeared reluctant to introduce Torres.
The former Inter Milan boss insisted his selection was down to rotation, but Cech knows, regardless of who plays up front, Chelsea must improve in front of goal to ensure they do not allow opponents back into games.
"It was disappointing because we didn't score against Swansea. We knew that we had to score to have a chance to progress, but we didn't and it is a pity," the 30-year-old said.
"Even when we played our game in hand in the league against Southampton, instead of killing the game off we played on and on and they managed to score two goals.
"I think we need to be more efficient in front of goal because we are having chances.
"In the last few games we haven't killed the games off. Against Arsenal it was the same story.
"We had to fight until the end of the game because we didn't take our chances to kill off the game. We need to be more clinical in front of goal."
Chelsea will look to make amends when they travel to face Brentford in the FA Cup on Sunday.
And Cech hopes to see a response from their League Cup disappointment in a competition Chelsea have won in four of the last six seasons.
He said: "We want to keep the record going and we are defending champions. We would like to defend the trophy."