This approach creates multiple sources of truth for how elements should look, and in general I would avoid it - in the same way that I avoid using the style attribute or <style> tags.
For structure (the meaning and content of each element), use HTML. For presentation (e.g. colors, fonts, transition appearance), stick to your linked stylesheet. If you need to modify behavior (e.g. what happens on a given event), do that in a Javascript include.
It really depends where you're coming from. For the kind of sites that prevailed maybe five years ago, you could separate HTML and CSS quite neatly and JS was a minor element used to add some effects. But these days you're starting to get web apps etc. with very rich interactivity; and once you're past a certain level of complexity it's inevitable that you'll end up modifying lots of HTML and CSS from JS. So if you reach that point then you can efficiently make JS your main (or even only) source of truth.
Ten years ago, we were all tempted to put <style> tags in the HTML "in order to have the entire website in one place." As site complexity grew, we realized we needed to start separating components that had distinct roles. If I'm changing a font site-wide, I don't want to scroll through lines of totally unrelated copy to get to my styles. I want completely separate files that only need to reference each other (via classes and ids).
I am a huge fan of AngularJS, because it's essentially following the same separation pattern. It uses references (like ng-model, ng-controller, etc.) to attach elements to associated interactive behavior, but doesn't actually interfere with the structure of the page.
Things like fonts can be defined at a high level by higher-level css tags or classes. I usually use less instead of direct css, so for me I'd just throw in an import in each view's less file to get the default styling applied, or I'd just have basic font styling done at a higher level in css so at the level closer to the actual class, I'd only need to apply tweaks as needed.
To be fair, so does Twitter Bootstrap, with its use of classes like "pagination-centered" - but even that's an antipattern. I tend to agree with etj's approach.
For structure (the meaning and content of each element), use HTML. For presentation (e.g. colors, fonts, transition appearance), stick to your linked stylesheet. If you need to modify behavior (e.g. what happens on a given event), do that in a Javascript include.