Protecting a 2-0 win from the first leg at Stamford Bridge, the Swans earned a goalless draw at the Liberty Stadium on Wednesday night to set up a showdown with League Two side Bradford City in the final.
"It's unbelievable tonight," Britton told Sky Sports News. "To do it over two legs against a team like Chelsea, the team have been brilliant in both games.
"Tonight we made history, which it great for Swansea. You wouldn't dream it."
Swansea's last piece of silverware came when they lifted the Football League Trophy in 2006.