A fixture against the Tigers back in May 2003 played a huge part in shaping the last decade of Swansea's history.
The Welsh club were on the brink of being relegated out of the Football League, and such an outcome would have made financial oblivion a real possibility.
But a James Thomas hat-trick gave Swansea a 4-2 win; a decade later they are a Premier League club and last season secured their first major silverware in the form of the Capital One Cup.
The sides have met since that day at Swansea's old Vetch Field home, but Britton, who played in the critical fixture, admits Monday's game will be that little more special.
He said: "We've played Hull a few times over the years, but every time it evokes the memories of that day because it's arguably the biggest and most important game in Swansea's history.
"We'll always have a special rapport with Hull because of that game. It will make Monday that much more special.
"It will be a great day for both clubs. Considering that game in 2003 and the fact 10 years ago we were both fighting in the bottom division, it's brilliant that both of us are playing in the Premier League on Monday night. It will be a special night.
"Anything is possible in football as long as you have the right people behind you. Swansea have had the right board, made the right decisions and had good players along the way. It shows you don't have to whack a lot of money into a football club to achieve success. We've done it in a different way."
And Britton has acknowledged that, while there is always pressure in the Premier League, this time round he will feel a lot more relaxed than when he stepped out onto the field back in 2003.
"I just remember the build-up to the game and how anxious everyone was," he said.
"But we had a dream start, went 1-0 up and the crowd was bouncing. Yet within 15-20 minutes we were 2-1 down and the atmosphere changed completely. You could hear a pin drop.
"That day was a range of emotions. The over-riding one was of relief when we got to 4-2 and the referee blew the final whistle.
"But the Premier League is a completely different pressure. Back then, you were talking about people paying for mortgages, players with families and no one knew what was going to happen.
"The majority of players - except one player - all had their contracts up so everyone was playing for their futures.
"If we'd lost that game no one knew what would happen.
"Financially, the players are well rewarded in the Premier League so you don't have to worry too much on that side of things.
"The pressure back then was a lot more intense with what was on the line for the players."