Translation:Last week I saw my husband's aunt.
Ver takes an indirect object here. The construction is "ver a alguien".
So, "La semana pasada vi a la tía de mi esposo" is a complete sentence.
What I think you might be confusing this with is previous sentences you've seen like "La semana pasada le vi a ella." The difference in this sentence is that the indirect object is a pronoun. The sentence starts off as "La semana pasada le vi", but this is ambiguous. It could mean "Last week I saw him/her/you (formal)". This is when you add the clarification to the end: 'a ella' 'a él' or 'a usted'.
I'm sorry Amy, but I think you've got this backward. (Since you have probably gotten this straightened out after four years, I'll still explain for the benefit of other users coming along.)
Ver uses the direct object, not the indirect. What is being seen = the aunt = direct object. That's why no DO pronoun la is used here. (In answer to MarkofSky's question.) Direct object pronouns are not used when the direct object noun is included in the sentence.
Without the noun, I saw her = La vi. Or La vi a ella, if the direct object pronoun la could be mistaken for it or you formal.
An indirect object receives the direct object (stated or implied). Here's the aunt as an indirect object: I sent my aunt a letter. Le envié a mi tía una carta. The stated DO is the letter, and the aunt is the IO, the receiver. Even though the aunt is used in the sentence, since she's the indirect object we need an indirect object pronoun (le) before the verb (envié).
Hope this helps to clear it up.
When I made a typo, the text at the bottom of the screen said: ''You used the wrong word: Last week I saw my hubby's aunt''. Hubby ? Apart from the fact that it's a ridiculously twee pet-name, like ''my doggy'', it doesn't match the translation at the top of this column. How can I believe anything DL tells me?