Can the belief of Jesus' listeners be considered belief unto salvation?

Belief in Jesus, God's Only Son, so that one might have eternal life [= achieve salvation], is the reason in scripture that God sent his Only Son, showing his love for us precisely by sending his Only Son.

cf. the famous and oft quoted John 3:16 (RSVCE):

16 For God so loved the world that he gave his only Son, that whoever believes in him should not perish but have eternal life.

Therefore the many that believed in him appeared to have arrived.

Please see also the return results on the biblical search: believe in God in the Gospel of John.

I choose another that resonates:

John 9:35-38 (RSVCE)
Spiritual Blindness 35 Jesus heard that they had cast him out, and having found him he said, “Do you believe in the Son of man?”[a] 36 He answered, “And who is he, sir, that I may believe in him?” 37 Jesus said to him, “You have seen him, and it is he who speaks to you.” 38 He said, “Lord, I believe”; and he worshiped him.

Footnotes:
a. John 9:35 Other ancient authorities read the Son of God.

Is there any biblical, or even logical, argument to conclude that this belief was unto salvation

For Catholics, to be saved, one needs to worship God by the theological virtues of Faith [= believe in God and in all he has revealed], Hope, and Charity. [cf. Penny Catechism, 8].

Please see also the Credo Chart.

Yes, I think it was a "belief unto salvation" and I think there is both a Biblical and logical argument to back that up.

Biblical

Looking at the Strong's words for this chapter, this particular word, pisteuw, means:

believe, commit unto

The root word, pistiv, is even more explicit:

conviction of the truth of anything, belief; in the NT of a conviction or belief respecting man's relationship to God and divine things, generally with the included idea of trust and holy fervour born of faith and joined with it

relating to God

the conviction that God exists and is the creator and ruler of all things, the provider and bestower of eternal salvation through Christ

and so on (see links for full details).

Logical

What else could "believe in Him" mean in this context? He had just said (v12):

"I am the light of the world. Whoever follows me will not walk in darkness, but will have the light of life."

and (v24):

"I told you that you would die in your sins, for unless you believe that I am he you will die in your sins."

So I cannot think of any other interpretation.

However, there is in addition a third reason:

Purpose of John's Gospel

Let's remember why John wrote his Gospel, explained in John 20:30-31:

Now Jesus did many other signs in the presence of the disciples, which are not written in this book; but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.

Since John wanted people to believe (or go on believing) in Jesus, it makes sense that this story backs up that purpose.

Can the belief of Jesus' listeners be considered belief unto salvation?

YES

Galatians 3:2-7 (NLT)

Let me ask you this one question: Did you receive the Holy Spirit by obeying the law of Moses? Of course not! You received the Spirit because you believed the message you heard about Christ. How foolish can you be? After starting your Christian lives in the Spirit, why are you now trying to become perfect by your own human effort? Have you experienced so much for nothing? Surely it was not in vain, was it? I ask you again, does God give you the Holy Spirit and work miracles among you because you obey the law? Of course not! It is because you believe the message you heard about Christ. In the same way, “Abraham believed God, and God counted him as righteous because of his faith.” The real children of Abraham, then, are those who put their faith in God.

Abraham was counted as righteous because of his faith. In the same manner, those who believed in the words of Jesus are cleansed through the Word of God.

You are already clean because of the word I have spoken to you. (John 15:3, NIV)

In passage you posted(John 8:30), Jesus was telling to the people that He is the Light of the world and the promised savior, who will save us from our sins. All they had to do was believe in Him and the Father who sent Him, and they will not die in their sins.

John 8: 24-30 (ESV)

I told you that you would die in your sins, for unless you believe that I am he you will die in your sins.” So they said to him, “Who are you?” Jesus said to them, “Just what I have been telling you from the beginning. I have much to say about you and much to judge, but he who sent me is true, and I declare to the world what I have heard from him.” They did not understand that he had been speaking to them about the Father. So Jesus said to them, “When you have lifted up the Son of Man, then you will know that I am he, and that I do nothing on my own authority, but speak just as the Father taught me. And he who sent me is with me. He has not left me alone, for I always do the things that are pleasing to him.” As he was saying these things, many believed in him.

Though Jesus was not yet crucified, faith in Him was sufficient to have salvation. For, The death of Jesus saves all the believers before and after His crucifixion.