The Potters fell behind on 34 minutes when Wilfried Bony converted from the spot, although Charlie Adam tucked away a penalty of his own shortly after, with Jonathan Walters eventually securing all three points for the home side.
Swansea managed to work a couple of half-chances late on, although Stoke stood firm to see out the game.
Speaking after the game, Shawcross told BBC Sport: "Swansea are a good side, they gave us problems in the first half, in the second half we switched it round and were very compact. Once we got the second goal we felt pretty comfortable and deserved the victory."
Stoke have climbed up to 10th place in the Premier League table with the win.