You have used through (definition 5, by means of) correctly in your sentence. Good job.
I will say that it is not all that natural. In other words, it's grammatical and understandable, but not idiomatic, not the most natural way to express the thought.
Through is often used as a result of a middleman, or to signify a process. Neither of these apply to your tips.
Also, are you thanking Alex directly? If so, you need commas before and after Alex:
Thanks, Alex, for all the money saved through your tips.
In addition, tips is ambiguous without further context, as the word can mean either (monetary) tips gained from working as a waiter, for example, or tips as in suggestions. I assume you mean the latter, and in any case it does not much affect the choice of preposition.
Changing no word except through, the following are perhaps preferable:
Thanks, Alex, for all the money saved from your tips.
Thanks, Alex, for all the money saved by your tips.
But, frankly, two more natural sentences are
Thanks, Alex, for all the money you (have) saved me by (means of) your tips.
Thanks, Alex, for all the money I (have) saved from/by (using/following) your tips.
Just as many native speakers would not normally choose the preposition through in either of these two sentences, we wouldn't normally choose it in your original sentence. There is no explanation as to Why, rather it's simply a matter of being familiar with which preposition sounds better, or "works" more naturally in a particular sentence.
I am a native speaker of American English.