Is it correct to say "Rationality has taken over"

I believe that your question what is the common way? is more profound than it seems, and that it deserves a detailed answer.

The phrase rationality took over could be the best answer, it could be an indifferent answer, or it could be a bad answer and you definitely want to use something else. It depends on where you are, and what effect you want to achieve.

Questions of style are not like questions of grammar and vocabulary, which often have a right or wrong answer (though not always). Style like wearing color combinations: it's a matter of taste and propriety. Even if most people may like it one way (convention), some individuals may have a "personal style" that works perfectly well for them. And those individuals (if male) might still wear black wherever "dinner jacket" is required.

As @Cascabel correctly pointed out, "to take over" sounds a little unconventional. There is a precise reason: that's because "to take over" often has a connotation of acting forcefully, as in "the army took over". And opposing "force and reason" is quite commonplace in our culture. Hence putting force and reason side by side is a "clash of colors" that could be frowned upon.

But it could also be praised as an oxymoron (a figure of speech where you clash two things on purpose); as in a deafening silence, or (indeed) peace broke out (which I personally love). There is a whole litterary genre called surrealism dedicated to practicing such incongruous juxtapositions.

So (in the extreme) the common way if you were a surrealist might indeed be to write rationality took over.

Now, if you want to use the common way in educated circles such as universities or the management of international corporations etc., you might want to write

Rationality prevailed and I gave up once and for all

(and that might indeed be what you are looking for?). That's because to prevail has a connotation of persuasion that goes well with rationality. Furthermore, prevail is quite dignified, and invites support from well-to-do people who dislike the excesses of common folk.

Indeed, would it be common , if you went to a shop, to hear a patron casually saying rationality prevailed? Unless it was some upscale boutique in London or New York, I wouldn't think so.

As @Cascabel wisely pointed out, you'd rather hear:

I came to my senses and gave it up for good.

Then the little details, such as using gave it up, rather than gave up are quirks that depend on the location, and who is talking, etc. If you really want to nail it (as when you are starting in a new city or on a new job) I'm afraid the only solution is to collect samples of what people are actually writing, or to listen carefully to what they are actually saying.