τῇ γὰρ χάριτί ἐστε σεσῳσμένοι διὰ πίστεως
διὰ is a preposition which is, technically, ad "verbal adjective." Participles usually introduce participial phrases which can serve either adverbially or adjectivally ("Running for your life." vs. "Hair of white.") and, as such, can take on a wide variety of meanings. To further widen the range, they can also be employed lexically or idiomatically. διὰ typically denotes "through" which would be locative but it also can be used instrumentally ("by"). In the case system, both of these categories are typically most naturally associated with the dative case. However, instrumentality can also be established with adverbs and adverbial phrases.
As with most of these sorts of studies, we're going to want to look at context. πίστεως is the next word, but how is it related? It's feminine, singular, genitive which may seem a bit paradigm-breaking but πίστεως can also be the object of a preposition, which is likely the usage here. It's an adverbial prepositional phrase. But in what sense does this modify the main verb (σεσῳσμένοι, [perfect, passive, part., masc. nom. pl.], "you all have come be be saved")?
διὰ is helpful because it does restrict the range of uses and that it is an adverbial phrase is also helpful. "By" and "through" being our standard candidates, let's look at them. But before we do ...
... the whole clause begins with τῇ ... χάριτί which is the articular dative of "grace." In English we have to convert this to a prepositional phrase ("in/by/with/through/into/etc. grace") in order to draw out the dative. This, then, becomes where the theology (and my Ordo salutis comment) comes into play because we now have what appear to be two instrumentals that are modifying the main phrase. The main part of this this whole clause (and it is an English clause all wrapped into one Koine word) is:
σεσῳσμένοι - "You have come to be saved."
This is modified by both
τῇ ... χάριτί - "in/by/with/through/into/etc. grace"
It may be helpful to diagram it so ...
τῇ ... χάριτί
As mentioned above, the translators now have a decision to make. Instrumentality candidates are means/agency, and manner. Given the simple action "The ball was hit.", means would define the action further by answering "how" ("The ball was hit by the batter.") while manner describes the action ("The ball was hit by the batter into the sky."). Agency is a personal use of means (which "the batter" could be considered personal so this example could be agency).
So how do these phrases relate to each other as they modify the main clause? Are they held in apposition (either both are manner or both are means)? Is one manner and the other means? Apposition can be ruled out because χάριτί and πίστεως are distinct concepts.
With all of this background in mind, we can look at the decision made by the translation committee. It appears as though they choose "by grace" as an instrumental of means, and "through faith" as instrumental of manner. I'd not spend too much time focusing on what διὰ actually means as far as verbal aspect (when in time) is concerned. Koine usually uses the verb tenses to highlight the aspect.