"As" is in all circumstances needed because it is the verb "avoir" in the second person singular used here as the necessary auxiliary for the tense, this tense being "passé composé". You may nevertheless ask what is "le" (l’) doing there.
This is a case of "dislocation à droite" (Wiktionnaire).

• (Grammaire) Déplacement d’un syntagme d’information additionnelle à la fin de la phrase, négligeant la structure ordinaire définie par le verbe, pour un rappel ou pour éviter une ambigüité. Par exemple, dans les phrases suivantes en français parlé, les syntagmes en lettres grasses sont placés à la fin par la dislocation à droite :
  a. C’est très amusant, ce film.
  b. Je l’ai vue au cinéma, Jeanne.

Translation – A noun phrase carrying additional information is displaced at the end of the sentence, thereby introducing a difference from the regular grammatical struture, and this is done so as to provide a reminder or avoid ambiguity. For instance, in the following sentences the noun phrases in bold type are cases of this dislocation.

There is a redundancy as "l’" stands for "pantalon". As said in the explanation above dislocation should be used only for essentially two reasons (reminder, avoiding ambiguity); however, if you parse enough spoken language you’ll find rapidly that a lot of people use it for no apparent reason and that forms such as this one that you are inquiring about have become a colloquial usage which is strictly equivalent to the normal form, here, "Où as-tu acheté ce pantalon ?". This latter form has become rather formal and is not used much in the spoken language. You can, however, still use it ; there is not yet the least obligation to use the alternative you mention. Let’s not forget the form "Où est-ce que tu as acheté ce pantalon ?". It is a little more complicated but used more often than "Où as-tu… ?" (ngram). Let’s remark as well that you can use dislocation with the basic form and also this latter one. (see correction in user Reyedi‘s comment)

• Où est-ce que tu l‘as acheté, ce pantalon ?
• Où l’as-tu acheté ce pantalon ?

"Tu as acheté où ce pantalon?" is not correct; it is not said at all. You might say though "Tu as acheté ce pantalon où ?" especially if you are asking with great insistence, case in which "où" is heavily stressed.

Technically, the sentence is missing a comma:

Tu l’as acheté où, ce pantalon ?

To parse it, better to first ignore the trailing part that is optional. That reads:

Tu l’as acheté où ? or the variants où tu l’as acheté ? and où l’as-tu acheté ? (formal)

This clearly translates to

Where did you buy it?

But the person speaking wants to make sure you know what the pronoun le refers to, so (s)he states it clearly either at the beginning or at the end of the sentence. That gives:

Ce pantalon, tu l’as acheté où ?

or

Tu l’as acheté où, ce pantalon ?

Finally, Tu as acheté où ce pantalon ? is possible although not very common but you can certainly hear:

Où tu as acheté ce pantalon ? (common) also Où t’as acheté…

Tu as acheté ce pantalon où ? (common) also T’as acheté…

Où as-tu acheté ce pantalon ? (formal)

T’as acheté où ce pantalon ? (colloquial)

They all means the same thing as above.

To complete the previous answers.
As said, the "correct" sentence would need an additional comma to look like :

"Tu l’as acheté où, ce pantalon ?"

As stated by others, the 2nd part of the sentence is to remind what we are talking about, and that second part is optional.
So without the optional part the sentence would be :

"Tu l’as acheté où ?"

And here unlike the sentence with the optional part, the " l’ " is important, it’s to point what we are talking about, without the 2nd sentence part, "Tu l’as acheté où ?" would commonly be used as a follow up sentence, after a sentence who was already referring to that "pantalon".
So basically, the " l’ " is the "what" you are referring to when asking something in that sentence.