Does a strikeout count as a put out?

Yes. A catcher gets credited with the putout on a strikeout if the ball is caught cleanly. In the event of a dropped third strike, if the catcher tags the hitter, he again gets credited with the putout. If the third strike is dropped and the catcher must throw down to first, the catcher gets credited with an assist and the first baseman gets credited with the putout. In no case would the pitcher get credit for the putout or assist.

Source: MLB Official Rule 10.09b