Exactly which export option are you using, and what are you using to view the resulting HTML file? If I export to Dynamic HTML or HTML and view the files with Google Chrome 21.0.1180.82 on the Mac (10.6.8), the right thing happens.
My test string was "this isn't for real" and the relevant line in the output is:
Code:
<td class=" content level1 subcell">this isn't for real<br>
Is it possible you've got something putting in a different character than the standard apostrophe, and then viewing on something that doesn't understand Unicode?