The Blues took a first-half lead through John Terry, who then went off injured following an accidental collision with Luis Suarez, who later equalised for the Reds.
Carragher told Sky Sports 1: "I think first half Chelsea were on top but in the second we came out and great credit to the lads.
"We probably deserved a draw in the end, the way we battled in the second half."
It was Carragher's first start in the league this season and he said: "The Premier League is a step up (from the cups) with all its intensity and Chelsea is a difficult place to come.
"If I am picked I give everything; it is down to the manager, he is the boss."
When asked if not playing was frustrating, Carragher said: "Yes and no. I want to play but I understand. I am never going to cause a problem."
Reds team-mate Jose Enrique added: "Most important is the performance from the team. Suarez is an amazing player, a key player for us."
Liverpool boss Brendan Rodgers also believed changing the system at half-time resulted in a far more effective team after the break.
He told Sky Sports 1: "I thought in the first half we were too tentative. Second half our attitude in the game was much better and we are better when we play fast football.
"Coming here is a very difficult place. I thought it was a great point for us. You have to give credit to the players, they were fantastic."
Rodgers also hailed Carragher, adding: "He is phenomenal, it's an absolute pleasure to work with him. His preparation is first class, he is always preparing himself well.
"Today [Sunday] he was exceptional, he is a real model professional for a young player.
"With everything he has won and achieved he could easily come and knock on my door but he is old school, he respects authority."
Chelsea manager Roberto Di Matteo revealed the club did not know the extent of Terry's knee injury with the captain - returning after his racism ban - set for a scan tomorrow [Monday].
Di Matteo told Sky Sports 1: "He has a problem with his knee and will have an MRI scan tomorrow [Monday].
"At the moment we can't say, we'll have to wait for the scan which will tell us the extent of the injury. He is certainly a player that is influential for us. We will miss him."
Terry was consoled by Di Matteo before disappearing down the tunnel.
"I was concerned about his well-being," the Italian added.
"We just got him back into the team after the ban. It was great seeing him out there."
Di Matteo said: "He started the game very well, showing the qualities he has, and the experience and influence.
"It's a real shame he had to come off as he was having a very good game."
Chelsea badly missed Terry during his ban and they struggled defensively after he went off today [Sunday], allowing Suarez to snatch a draw.
To compound matters, Ashley Cole was already sidelined with a hamstring problem and David Luiz was out with tonsilitis.
It all added up to a potential defensive injury crisis for Chelsea.
"At centre-back we have cover," said Di Matteo, who was "hopeful" Luiz and Cole would return for Saturday's trip to West Brom.
Midfielder Frank Lampard was also still missing today with a calf problem.
Di Matteo added: "Lamps, it might be a bit too early, but he won't be long after that."
The Italian was frustrated by his side's inability to claim a second goal and ultimately allow the visitors back into the game.
"We controlled the game and had chances for a second goal and that is probably a disappointment, 1-0 is always dangerous," he added. "We needed to score a second one.
"He (Reds goalkeeper Brad Jones) made some very good saves and also some misses from us.
"We knew we needed a second goal and pushed it in the second half and we needed to take it."