The striker scored both goals in the first half, with slack defending by the visitors allowing him to head in Glenn Whelan's corner in the 12th minute.
Crouch then found little resistance in the Swansea box once again nine minutes before the break as he fired the ball at Michel Vorm and reacted to convert the rebound, taking his tally for the campaign to five goals in all competitions.
That form will keep the debate going about a potential England recall for the 31-year-old, while his club will perhaps be breathing a little easier now having finally recorded a victory in their sixth Premier League game of the season after drawing four and losing one of the opening five.
Swansea, meanwhile, started the campaign well but have now lost three league matches in a row and need to turn things around quickly.
Stoke began the contest brightly and grabbed an early lead after Vorm's save from Jon Walters led to a corner.
Crouch was left unmarked when it was swung in by Whelan, allowing the frontman to crash a header past the helpless Vorm.
Swansea were almost in trouble at the back again moments later as Vorm was charged down by Crouch, but they survived that before seeing Charlie Adam rifle an attempt over the bar.
The visitors looked to apply some pressure of their own and after Michu had struck a weak effort at Asmir Begovic, a lapse at the back by the Potters saw them pass the ball straight to Danny Graham, whose shot on the turn was saved.
At the other end Vorm, who had prepared for the match by training with rugby tackle bags, showed his strength by coming out to beat Adam to the ball and punch it away, a collision which left the Stoke man temporarily down hurt.
After getting back to his feet Adam sent a curler into Vorm's arms and Michu then bent an effort just the wrong side of Begovic's left post.
The hosts regained their momentum and in the 36th minute they doubled their advantage as Crouch struck once more.
Sloppiness by Swansea again played its part, with Crouch able to bring the ball down from a floated cross into the area and then tuck it away on the follow-up after Vorm had parried his initial shot.
The away side came out after the restart looking full of purpose and Begovic was called upon to tip a Ki Sung-yueng effort over the bar.
Ben Davies then went down in the box, seeming to look for a penalty and being rewarded only with a booking for diving - one of the hot topics for Tony Pulis in the build up to the game.
Swansea continued to attack and Graham might have done better with an effort he scooped over the goalframe.
Stoke hit back and Michael Kightly was just unable to convert from close range as he strived to meet Adam's cross.
Crouch looked intent on securing his hat-trick, but headed one effort on to the bar and then missed with an attempted overhead kick before being substituted.
Kightly and Walters also tried their luck towards the end to no avail, but the Potters had done enough.
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