The two teams meet for the first time in the Premier League - live on Monday Night Football - with the home side looking to move into the top half of the table with a win.
Britton is injured for tonight's game but told Sky Sports News about the game he played in 10 years ago - which took place at Swansea's former home, the Vetch Field.
"Those 10 years have gone really quick," he said. "But every time we play Hull, it always brings back the memories of 2003 when we had to win that game to stay in the football league.
"They're great memories - it was a very intense match and one that has gone down in the club's history.
"You only have to look at the stadium we played in to see where we were in 2003 and where we are now - the Premier League is worlds apart.
"It shows how far the football club has come in such a short time - it's been an incredible journey."
Britton also paid tribute to Monday night's opponents who have also transformed their fortunes to take their place among football's elite.
"Hull have also had a fantastic rise through the leagues - it's great to see. These are the kind of stories everyone wants to see - and Hull's story is a great one."
Swansea v Hull is live on Monday Night Football, Sky Sports 1HD at 7pm