I think I know what your looking for in an answer here. I say, and I believe it is the Scripture that teaches, that regeneration necessarily precedes conversion. Or as you asked, we are saved and believe.
I use the word regeneration and conversion on purpose. Regeneration means being born again. Conversion means repentance and faith in Christ Jesus. So, I say again, Regenereation necessarily precedes conversion.
There are several texts of Scripture that answer this question.
1 John 2:29
If ye know that he is righteous, ye know that every one that doeth righteousness is born of him.
1 John 3:29
Whosoever is born of God doth not commit sin; for his seed remaineth in him: and he cannot sin, because he is born of God.
1 John 4:7
Beloved, let us love one another: for love is of God; and every one that loveth is born of God, and knoweth God.
1 John 5:1
Whosoever believeth that Jesus is the Christ is born of God: and every one that loveth him that begat loveth him also that is begotten of him.
Now, the verb "born" is in the perfect tense showing that this action precedes the other actions, that of practicing righteousness, not committing sin, loving, and believing. (see resource)
I am sure we would all agree that the practicing of righteousness, loving the brethren, and not committing (habitual) sin, are things that follow, or come after being born again. The same is said of "whosoever believeth.."
It was the Lord who opened Lydia's heart so that she "attended unto the things which were spoken of Paul." Acts 16:14.
Jesus taught, "All that the Father giveth me shall come to me; and him that cometh to me I will in no wise cast out." John 6:37 It those whom the Father draws/gives to the Son that shall (not might/maybe/or possibly) come and believe on the Lord Jesus Christ.
Let us not get too hung up on timing. It may that the things happen simultaneously, but certainly not before. In this way no glory can be given to man, but all glory to God. Amen.
RB
resource:
http://www.9marks.org/CC/article/0,,PTID314526|CHID598016|CIID1731702,00.html