Eagles striker Alexander Sorloth, signed from Midtjylland in January, scored his first goal of the club late in the second half.
The Premier League club dominated the closing stages of the match and were unlucky not to extend their lead.
"I'm happy," said Roy Hodgson in his post-match press conference. "I thought that was a good game. In the first half Swansea were very good, they got around the ball really well, they were sharp, they were quick and they had a really good spell at the start of that first half.
"It took us a while, I thought, to get our game together. We were going too deep to receive the ball, we had too many players around the ball which meant that we were playing tip-tap football really and that suited Swansea because they got around us and made life difficult."