Yes, he was saved; according to the author of the Fourth Gospel, and going entirely by that gospel (written as it is by an eye witness) - I think that is what we are intended to infer: that Pilate 'believed on' the name of Jesus; and therefore was saved.
I feel that this is why so much attention is given to each of several interactions between Jesus and Pilate.
For me the crucial aspects seem to be firstly that Pilate asserted plainly and repeatedly (and would not withdraw the statement) that Jesus was King of the Jews, meaning he was the Messiah, the spiritual king (by contrast, the Jews said - "We have no king but Caesar").
Secondly that Jesus said to Pilate: "Thou couldest have no power at all against me, except it
were given thee from above: therefore he that delivered me unto thee
hath the greater sin." - which, I take it, absolves Pilate of direct responsibility for crucifying Jesus.
In short, we are shown that Pilate believed Jesus's claims concerning himself, and behaved as such - this (and this alone), we are repeatedly told, is what qualifies us for Life Eternal; and this is why Pilate is given such great prominence in the Fourth Gospel.
Note: Pilate comes out worse if the other Gospels are also taken into account; but the purpose of my current reading it to take the fullest account of the greatest authority of the Fourth Gospel. Here is the relevant text:
From John Chapters 18 & 19: Pilate then went out unto them, and said, What accusation bring ye against this man? They answered and said unto him, If he were not a malefactor, we would not have delivered him up unto thee. Then said Pilate unto them, Take ye him, and judge him according to your law. The Jews therefore said unto him, It is not lawful for us to put any man to death: That the saying of Jesus might be fulfilled, which he spake, signifying what death he should die. Then Pilate entered into the judgment hall again, and called Jesus, and said unto him, Art thou the King of the Jews? Jesus answered him, Sayest thou this thing of thyself, or did others tell it thee of me? Pilate answered, Am I a Jew? Thine own nation and the chief priests have delivered thee unto me: what hast thou done? Jesus answered, My kingdom is not of this world: if my kingdom were of this world, then would my servants fight, that I should not be delivered to the Jews: but now is my kingdom not from hence. Pilate therefore said unto him, Art thou a king then? Jesus answered, Thou sayest that I am a king. To this end was I born, and for this cause came I into the world, that I should bear witness unto the truth. Every one that is of the truth heareth my voice. Pilate saith unto him, What is truth? And when he had said this, he went out again unto the Jews, and saith unto them, I find in him no fault at all. But ye have a custom, that I should release unto you one at the passover: will ye therefore that I release unto you the King of the Jews? Then cried they all again, saying, Not this man, but Barabbas. Now Barabbas was a robber.Then Pilate therefore took Jesus, and scourged him. And the soldiers platted a crown of thorns, and put it on his head, and they put on him a purple robe, And said, Hail, King of the Jews! and they smote him with their hands. Pilate therefore went forth again, and saith unto them, Behold, I bring him forth to you, that ye may know that I find no fault in him. Then came Jesus forth, wearing the crown of thorns, and the purple robe. And Pilate saith unto them, Behold the man! When the chief priests therefore and officers saw him, they cried out, saying, Crucify him, crucify him. Pilate saith unto them, Take ye him, and crucify him: for I find no fault in him. The Jews answered him, We have a law, and by our law he ought to die, because he made himself the Son of God. When Pilate therefore heard that saying, he was the more afraid; And went again into the judgment hall, and saith unto Jesus, Whence art thou? But Jesus gave him no answer. Then saith Pilate unto him, Speakest thou not unto me? knowest thou not that I have power to crucify thee, and have power to release thee? Jesus answered, Thou couldest have no power at all against me, except it were given thee from above: therefore he that delivered me unto thee hath the greater sin. And from thenceforth Pilate sought to release him: but the Jews cried out, saying, If thou let this man go, thou art not Caesar's friend: whosoever maketh himself a king speaketh against Caesar. When Pilate therefore heard that saying, he brought Jesus forth, and sat down in the judgment seat in a place that is called the Pavement, but in the Hebrew, Gabbatha. And it was the preparation of the passover, and about the sixth hour: and he saith unto the Jews, Behold your King! But they cried out, Away with him, away with him, crucify him. Pilate saith unto them, Shall I crucify your King? The chief priests answered, We have no king but Caesar. Then delivered he him therefore unto them to be crucified. And they took Jesus, and led him away. And he bearing his cross went forth into a place called the place of a skull, which is called in the Hebrew Golgotha: Where they crucified him, and two other with him, on either side one, and Jesus in the midst. And Pilate wrote a title, and put it on the cross. And the writing was Jesus Of Nazareth The King Of The Jews. This title then read many of the Jews: for the place where Jesus was crucified was nigh to the city: and it was written in Hebrew, and Greek, and Latin. Then said the chief priests of the Jews to Pilate, Write not, The King of the Jews; but that he said, I am King of the Jews. Pilate answered, What I have written I have written.
Note: Pilate makes two more appearances after this, first to allow the the crucified men's legs to be broken and finish the process; and to allow Joseph of Arimathaea to take the body for entombment.