To paraphrase Igor Pak: OK, this I know. It is remarkable how difficult it is to track down a reference which gives an actual proof for this fact (moreover applicable to all global fields). The notes of Pete Clark don't give a proof or a reference for a proof, and its omission in Cassels-Frohlich is an uncorrected error. :)

But here is a reference: Theorem 3.6 in the notes on Honda-Tate theory on Kirsten Eisentraeger's webpage. The assertion is even stronger: one can find a *cyclic* splitting field of the expected minimal degree. A moment's reflection leads one to realize what is actually going on: in the non-archimedean local theory we know that one can always arrange the splitting field to be the *unramified* one of the expected minimal degree (already in Serre's Local Fields, and part of the story of the "local invariant"), so in particular it is cyclic in that case. Taking into account the real case, and using the exactness at the left of the global-to-local sequence for Brauer groups, the global problem reduces to making a global cyclic extension inducing specified local ones at finitely many places and having a predicted degree which is lcm of local degrees (in the local theory the degree is actually all that really matters, not the unramifiedness).

Enter Grunwald-Wang... and since all that matters locally is the degree, if we don't care about global cyclicity but just global degree and some local degrees then weak approximation & Krasner's Lemma suffice to do the job (so for the question as asked, in which there's no cyclicity, the global problem is actually very elementary once the local case is settled!). Note that in Cassels-Frohlich the global cyclic splitting field is addressed, but not its degree (since Grunwald-Wang is not addressed in Cassels-Frohlich).

Historically the existence of a global *cyclic* splitting field, moreover of the expected degree, was regarded as one of the real triumphs of global class field theory, and the early attempts at class field theory by the German school were intimately tied up with this problem of the cyclic splitting field. This is why it was such a shock to Artin when Wang discovered that Grunwald's proof of local-to-global for cyclic extensions was not true (but fortunately Wang's fix was sufficient); see Roquette's historical notes on CFT.

Finally, to put this in perspective, it should be noted (as remarked in Eisentraeger's notes) that there are examples of complex function fields in transcendence degree 3 admitting nontrivial 2-torsion Brauer classes not represented by a quaternion division algebra! (The appearance of trdeg 3 is reasonable, as the period-index problem for surfaces over an algebraically closed field was proved by deJong.)