The towering centre-forward opened the scoring in the 12th minute when he rammed home Glenn Whelan's inswinging corner with a customary header for his third goal of the season.
Crouch doubled the Potters' lead with a lucky goal ten minutes prior to half-time when Swansea failed to deal with a long ball leaving the England striker to smash one attempt at Michel Vorm before guiding the follow-up beyond the Dutchman.
Swansea's Michu fired wide just before the half-hour mark before Ki Sung-yeung produced a fine save from Stoke keeper Asmir Begovic.
Danny Graham was then unable to hit the target at the back post in the 66th minute as the Swans looked to get back into the contest which could have been put away if for Crouch's 78th minute header which flicked the crossbar.
Britannia Stadium, Stoke-on-Trent - 27,330