The 29-year-old is expected to be sidelined for up to six months after breaking his leg in a golf buggy accident in late August, but he has inked a new four-year agreement.
"I'm delighted to sign the contract because it shows the club have a lot of faith in me," Tate told the club's official website.
"It will take me well into my 30s and who knows what will happen beyond that, although obviously I would like to stay even longer.
"It will depend on injuries and form, but I would love to finish my career at Swansea and, with a bit of luck, I hope that could happen.
"We've been talking about a contract for a while. The injury delayed things a bit. I was meant to sign before I got injured, but then it got put on the backburner.
"Then it was deadline day, which held things up again, but the new contract was never under threat because of the injury.
"The club could see it was an unfortunate accident which no-one could have expected."