Long-serving midfielder Britton gave up the captaincy to join Paul Clement's coaching staff after Claude Makelele took over at Belgian club KAS Eupen, leaving the club searching for a new skipper.
That honour has been given to Spanish full-back Rangel, who has made 370 appearances for the Swans since joining the club from Terrassa a decade ago.
Speaking to the club's official website, the 35-year-old expressed his delight at being named club captain, though he admitted that it came as something of a shock.
"I never thought I would end up as a captain, but I am proud and honoured and full of enthusiasm after being given this chance," he explained.
"I thanked the manager. He could have chosen someone who would play a bit more regularly than me, but it is not just on the pitch that you need some leadership and I am happy to take this job.
"I had a lot of dreams to achieve when I first came here, one of them was to become a Premier League player, which I achieved.
"I've got Swansea in my heart. I love the club and the most important thing is that we stay in the Premier League this year."
Despite the player's surprise, manager Celemnt was in no doubt that the defender has all the necessary qualities for the role.
"Given his age, experience and the excellent service he has given to this club over a number of years, Angel is the right choice to be club captain," he said.
"He is a great role model and a very good professional who all other players can look up to."
The Welsh side are currently 19th in the Premier League table after having won just one of their last eight games, but the Spaniard insists that the club can turn their form around if they stick together.
"I think when we are going through a tough spell like we are at the moment, it's important that everyone sticks together and that starts with me," he pointed out.
"I can bring experience, some leadership alongside other players, a lot of team spirit I hope.
"The team spirit is one of the most important things in football. I think we have it here and I want to keep that going or even improve it if I can."