The game finished 2-2 after extra-time at Wembley before both Gerrards missed spot-kicks, but it was Anthony's which proved crucial in sudden death.
Steven Gerrard told BBC1: "It's always cruel when it comes to penalties, but there has to be a loser.
"It was always going to be that one of us was sad and one celebrating. I've got mixed emotions. Obviously I'm delighted to win the cup for our supporters, but I feel for Anthony and Cardiff.
"It doesn't matter what I say to him - he's going to be down. I'll be there for him and all the family will be behind him."
Substitute Dirk Kuyt thought he had won it in the 108th minute when he put the Merseysiders 2-1 up, but Ben Turner forced penalties with just two minutes remaining.
"It was incredible, an incredible day," Kuyt said. "I was very disappointed not to play longer than 15 minutes but I have to put that aside. I'm so happy to have scored the goal.
"The most important thing was we kept believing and in the end I think we deserved this trophy."
Joe Mason earlier opened the scoring for the Bluebirds in the first half only for Martin Skrtel to equalise on the hour mark.
Kuyt, who was one of Liverpool's three players on target from the spot, continued: "We keep believing. I said to Steven, there's a few more to take and others can miss and they did.
"I knew I had to score my penalty or else it was all over and that's what I did. I have lots of respect for the manager, the team, the supporters. They deserved more but we wanted this trophy so desperately."
It was a first medal in six years with the club for Kuyt, who has now set his sights on a double with Liverpool still involved in the FA Cup.
"This is why I came to Anfield," he added. "To finally get my first medal at Wembley is great. Hopefully we can get one more later in the season."
Skrtel, who brought Liverpool level at 1-1, hoped the Carling Cup would be followed by many more trophies.
"It was a great game in a beautiful stadium with an unbelievable atmosphere," the defender said on Sky Sports 1. "I think we deserved it. It wasn't easy because they equalised with two minutes to go, but we just tried to keep our heads up and we got it on penalties.
"I think everyone plays football for the trophies. This is the first one and I hope there will be more and more."
Cardiff boss Malky Mackay was beaming with pride despite his side's defeat.
"In extra time the way we came back... I'm so proud of them, so very, very proud of them," he said. "For most of them it's the first time they've been anywhere near something like this. They are very young. The emotions are running through them.
"You've got two teams who have put everything into the game today and I'm very proud of my team."