Goals from Jason Puncheon and former Bluebird Joe Ledley left the Welsh capital side staring down the barrel of relegation.
Ole Gunnar Solskjaer's side, who won the Championship last term to earn a place in the top flight, sit six points adrift of safety, with just five games to play.
Caulker admits Cardiff's performance in south Wales on Saturday was not up to scratch and called on his team-mates to discover a higher level of fight and passion if they are to survive the drop.
"It's difficult to explain it," the Bluebirds skipper told the Daily Star. "I don't know what went wrong.
"It was not for a lack of trying because the boys ran their socks off. But we are going to have to dig deep and show a lot more fight and passion.
"We apologise to the fans that turned up and gave us great support as always. We were very disappointed with it. We know we can perform a lot better than that."
Defeat leaves Cardiff 19th in the Premier League table, just one point above bottom club Sunderland, although the Black Cats hold three games in hand.
Cardiff travel to St Mary's on Saturday to face mid-table Southampton, before a home tie with Stoke and a trip to the Stadium of Light to take on Sunderland.
Relegation is not yet a certainty for Solskjaer's side but Caulker insists they have left themselves facing an uphill battle.
"We've given ourselves a mountain to climb now," he admitted. "It's going to be five cup finals between now and the end of the season.
"It's still mathematically possible we can stay up, so we are not going to write ourselves off.
"We've got confidence from how we played at the start of the season and we've got to get back into some kind of form."