Wednesday's 2-0 win over Chelsea, which came courtesy of goals from Michu and Danny Graham, means Swansea are now red-hot favourites to reach their first major Wembley final in history.
Should the Swans see out the second leg at the Liberty Stadium - where they have lost just twice this season - it will provide another chapter in what has been a remarkable decade or so for the club.
Saved from financial oblivion by a local consortium in 2002, the Welsh side survived relegation from League Two on the last day of the following season, but from there it has been a success story, with the club climbing the leagues to reach the heady heights they now occupy.
Having joined the club 10 years ago, Britton has been with Swansea almost every step of the way and the midfielder on Wednesday struggled to sum up his emotions after the historic win in west London.
"It's hard to describe how I'm feeling," Britton said.
"It's up there as 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.
"Remember, 10 years ago this May we were beating Hull to stay in the Football League."
Swansea enjoyed a memorable return to the top flight under Brendan Rodgers last season, recording home wins over Liverpool, Arsenal and Manchester City.
It was anticipated that the Swans may have struggled following Rodgers' departure, but with the addition of players like Michu, Chico and Pablo Hernandez, Michael Laudrup has taken the Welsh club to the next level this term.
"Last season we did very well, but we didn't beat one of the big teams on the road," Britton said on Swansea's website.
"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."
Yet before departing on the three-and-a-half hour coach trip back to south Wales, Britton was keen to sound a warning to his team-mates and the Swansea fans who are already dreaming of going to Wembley on February 24.
"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."
Although Chelsea will rely on the likes of Eden Hazard, Oscar and Juan Mata to deliver a surprise comeback, the Swans know they have a more lethal weapon in their own arsenal.
Michu, a £2million signing from Rayo Vallecano last summer, now has 16 goals to his name in all competitions.
Last night's strike, which came after the first of two glaring errors from Branislav Ivanovic, may prove to be the most important of his career at the Liberty Stadium, however.
Britton has been so impressed with Michu that he rates the 26-year-old up there with £24m Manchester United marksman Robin van Persie.
Britton said: "For £2million I think, although Robin van Persie might be the signing of the season, you've got to say Michu is right up alongside him.
"The chance he had was still a tough one to put away but he's done it all season."