Michu and Danny Graham scored a goal in each half to give the Swans a 2-0 victory at Stamford Bridge and put them in a great position to reach a first Wembley final in their history.
And Britton, who has represented Swansea in all four English divisions, said: "It's hard to describe how I'm feeling. It's up there with the best performances and results since I've been at the club.
"To go to the European champions, score two goals, and not concede any with the attacking quality they have got, is a great achievement for this club. For me, this result is right up there among the highlights of my career.
"Remember, 10 years ago this May we were beating Hull to stay in the Football League."
After a successful debut season in the Premier League last term, it had been feared that the departure of manager Brendan Rodgers might presage a tough 2012/13 campaign for Swansea.
But they have gone from strength to strength under Michael Laudrup and Britton added: "Last season we did very well but we didn't beat one of the big teams on the road.
"This season, though, we have gone to Liverpool, Arsenal, Newcastle and Chelsea and won. It gives you great confidence, and that is where we have improved on last season.
"You need guts, determination and bravery to go to a place like Stamford Bridge and win. We showed those qualities in abundance. The passing was good but everyone stood up and dug deep."
However, Britton says Swansea cannot afford to relax in the second leg of the semi-final at the Liberty Stadium on 23 January.
"We are halfway through this tie," he said. "I know it's boring but we can't get complacent because we have 90 minutes to play at home against a team of Chelsea's quality."